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Title: Mysticism and Logic and Other Essays

Author: Bertrand Russell

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The link to the Index has been added for the benefit of easy access.Obvious typographical errors have been corrected. For a complete list, pleasesee the end of this document. BERTRAND RUSSELL

MYSTICISM AND LOGIC

AND OTHER ESSAYS

LONDON GEORGE ALLEN & UNWIN LTD RUSKIN HOUSE MUSEUM STREET

MYSTICISM AND LOGIC

AND OTHER ESSAYS BY BERTRAND RUSSELL

The ABC of Relativity

The Analysis of Matter Human Society in Ethics and Politics The Impact of Science on Society New Hopes for a Changing World Authority and the Individual Human Knowledge History of Western Philosophy The Principles of MathematicsIntroduction to Mathematical Philosophy The Analysis of Mind Our Knowledge of the External World An Outline of Philosophy The Philosophy of Leibniz An Inquiry into Meaning and Truth Logic and Knowledge The Problems of Philosophy Principia Mathematica Common Sense and Nuclear Warfare Why I am Not a Christian Portraits from Memory My Philosophical Development Unpopular Essays Power In Praise of Idleness The Conquest of Happiness Sceptical Essays The Scientific Outlook Marriage and Morals Education and the Social Order On Education Freedom and Organization Principles of Social Reconstruction Roads to Freedom Practice and Theory of Bolshevism Satan in The Suburbs Nightmares of Eminent Persons

This book is copyright under the Berne Convention. Apart from any fairdealing for the purpose of private study, research, criticism or review, aspermitted under the Copyright Act, 1956, no portion may be reproduced by anyprocess without written permission. Enquiry should be made to the publisher. PRINTED IN GREAT BRITAIN by Taylor Garnett Evans & Co. Ltd., Watford, Herts.

[v] PREFACE

The following essays have been written and published at various times, andmy thanks are due to the previous publishers for the permission to reprint them. The essay on "Mysticism and Logic" appeared in the Hibbert Journal forJuly, 1914. "The Place of Science in a Liberal Education" appeared in twonumbers of The New Statesman, May 24 and 31, 1913. "The Free Man'sWorship" and "The Study of Mathematics" were included in a formercollection (now out of print),Philosophical Essays, also published by Messrs.Longmans, Green & Co. Both were written in 1902; the first appearedoriginally in the Independent Review for 1903, the second in the NewQuarterly, November, 1907. In theoretical Ethics, the position advocated in"The Free Man's Worship" is not quite identical with that which I hold now: Ifeel less convinced than I did then of the objectivity of good and evil. But thegeneral attitude towards life which is suggested in that essay still seems to me,in the main, the one which must be adopted in times of stress and difficulty bythose who have no dogmatic religious beliefs, if inward defeat is to be avoided. The essay on "Mathematics and the Metaphysicians" was written in 1901,and appeared in an American magazine, The International Monthly, under thetitle "Recent Work in the Philosophy of Mathematics." Some points [vi]in thisessay require modification in view of later work. These are indicated infootnotes. Its tone is partly explained by the fact that the editor begged me tomake the article "as romantic as possible." All the above essays are entirely popular, but those that follow are somewhatmore technical. "On Scientific Method in Philosophy" was the Herbert Spencerlecture at Oxford in 1914, and was published by the Clarendon Press, whichhas kindly allowed me to include it in this collection. "The UltimateConstituents of Matter" was an address to the Manchester PhilosophicalSociety, early in 1915, and was published in the Monist in July of that year. Theessay on "The Relation of Sense-data to Physics" was written in January, 1914,and first appeared in No. 4 of that year's volume of Scientia, an InternationalReview of Scientific Synthesis, edited by M. Eugenio Rignano, publishedmonthly by Messrs. Williams and Norgate, London, Nicola Zanichelli,Bologna, and Félix Alcan, Paris. The essay "On the Notion of Cause" was thepresidential address to the Aristotelian Society in November, 1912, and waspublished in their Proceedings for 1912-13. "Knowledge by Acquaintance andKnowledge by Description" was also a paper read before the AristotelianSociety, and published in their Proceedings for 1910-11. LONDON,September, 1917

[vii]

CONTENTS

Chapter Page I. Mysticism and Logic 1 The Place of Science in a Liberal II. 33 Education III. A Free Man's Worship 46 IV. The Study of Mathematics 58 V. Mathematics and the Metaphysicians 74 VI. On Scientific Method in Philosophy 97 VII. The Ultimate Constituents of Matter 125 VIII. The Relation of Sense-data to Physics 145 IX. On the Notion of Cause 180 Knowledge by Acquaintance and X. 209 Knowledge by Description Index 233

[1] MYSTICISM AND LOGIC AND OTHER ESSAYS

IToC

MYSTICISM AND LOGIC

Metaphysics, or the attempt to conceive the world as a whole by means of

thought, has been developed, from the first, by the union and conflict of twovery different human impulses, the one urging men towards mysticism, theother urging them towards science. Some men have achieved greatness throughone of these impulses alone, others through the other alone: in Hume, forexample, the scientific impulse reigns quite unchecked, while in Blake a stronghostility to science co-exists with profound mystic insight. But the greatest menwho have been philosophers have felt the need both of science and ofmysticism: the attempt to harmonise the two was what made their life, and whatalways must, for all its arduous uncertainty, make philosophy, to some minds, agreater thing than either science or religion. Before attempting an explicit characterisation of the scientific and themystical impulses, I will illustrate them by examples from two philosopherswhose greatness lies in the very intimate blending which they achieved. Thetwo philosophers I mean are Heraclitus and Plato. [2]Heraclitus,as every one knows, was a believer in universal flux: timebuilds and destroys all things. From the few fragments that remain, it is noteasy to discover how he arrived at his opinions, but there are some sayings thatstrongly suggest scientific observation as the source. "The things that can be seen, heard, and learned," he says, "are what I prizethe most." This is the language of the empiricist, to whom observation is thesole guarantee of truth. "The sun is new every day," is another fragment; andthis opinion, in spite of its paradoxical character, is obviously inspired byscientific reflection, and no doubt seemed to him to obviate the difficulty ofunderstanding how the sun can work its way underground from west to eastduring the night. Actual observation must also have suggested to him hiscentral doctrine, that Fire is the one permanent substance, of which all visiblethings are passing phases. In combustion we see things change utterly, whiletheir flame and heat rise up into the air and vanish. "This world, which is the same for all," he says, "no one of gods or men hasmade; but it was ever, is now, and ever shall be, an ever-living Fire, withmeasures kindling, and measures going out." "The transformations of Fire are, first of all, sea; and half of the sea is earth,half whirlwind." This theory, though no longer one which science can accept, is neverthelessscientific in spirit. Science, too, might have inspired the famous saying towhich Plato alludes: "You cannot step twice into the same rivers; for freshwaters are ever flowing in upon you." But we find also another statementamong the extant fragments: "We step and do not step into the same rivers; weare and are not." [3]The comparison of this statement, which is mystical, with the one quotedby Plato, which is scientific, shows how intimately the two tendencies areblended in the system of Heraclitus. Mysticism is, in essence, little more than acertain intensity and depth of feeling in regard to what is believed about theuniverse; and this kind of feeling leads Heraclitus, on the basis of his science,to strangely poignant sayings concerning life and the world, such as: "Time is a child playing draughts, the kingly power is a child's." It is poetic imagination, not science, which presents Time as despotic lord ofthe world, with all the irresponsible frivolity of a child. It is mysticism, too,which leads Heraclitus to assert the identity of opposites: "Good and ill areone," he says; and again: "To God all things are fair and good and right, butmen hold some things wrong and some right." Much of mysticism underlies the ethics of Heraclitus. It is true that ascientific determinism alone might have inspired the statement: "Man'scharacter is his fate"; but only a mystic would have said: "Every beast is driven to the pasture with blows"; and again: "It is hard to fight with one's heart's desire. Whatever it wishes to get, itpurchases at the cost of soul"; and again: "Wisdom is one thing. It is to know the thought by which all things aresteered through all things."[1] Examples might be multiplied, but those that have been given are enough toshow the character of the man: the facts of science, as they appeared to him, fedthe [4]flame in his soul, and in its light he saw into the depths of the world bythe reflection of his own dancing swiftly penetrating fire. In such a nature wesee the true union of the mystic and the man of science—the highest eminence,as I think, that it is possible to achieve in the world of thought. In Plato, the same twofold impulse exists, though the mystic impulse isdistinctly the stronger of the two, and secures ultimate victory whenever theconflict is sharp. His description of the cave is the classical statement of beliefin a knowledge and reality truer and more real than that of the senses: "Imagine[2] a number of men living in an underground cavernous chamber,with an entrance open to the light, extending along the entire length of thecavern, in which they have been confined, from their childhood, with their legsand necks so shackled that they are obliged to sit still and look straightforwards, because their chains render it impossible for them to turn their headsround: and imagine a bright fire burning some way off, above and behind them,and an elevated roadway passing between the fire and the prisoners, with a lowwall built along it, like the screens which conjurors put up in front of theiraudience, and above which they exhibit their wonders. I have it, he replied. Also figure to yourself a number of persons walking behind this wall, andcarrying with them statues of men, and images of other animals, wrought inwood and stone and all kinds of materials, together with various other articles,which overtop the wall; and, as you might expect, let some of the passers-by betalking, and others silent. [5]You are describing a strange scene, and strange prisoners. They resemble us, I replied. Now consider what would happen if the course of nature brought them arelease from their fetters, and a remedy for their foolishness, in the followingmanner. Let us suppose that one of them has been released, and compelledsuddenly to stand up, and turn his neck round and walk with open eyes towardsthe light; and let us suppose that he goes through all these actions with pain,and that the dazzling splendour renders him incapable of discerning thoseobjects of which he used formerly to see the shadows. What answer should youexpect him to make, if some one were to tell him that in those days he waswatching foolish phantoms, but that now he is somewhat nearer to reality, andis turned towards things more real, and sees more correctly; above all, if hewere to point out to him the several objects that are passing by, and questionhim, and compel him to answer what they are? Should you not expect him to bepuzzled, and to regard his old visions as truer than the objects now forced uponhis notice? Yes, much truer.... Hence, I suppose, habit will be necessary to enable him to perceive objects inthat upper world. At first he will be most successful in distinguishing shadows;then he will discern the reflections of men and other things in water, andafterwards the realities; and after this he will raise his eyes to encounter thelight of the moon and stars, finding it less difficult to study the heavenly bodiesand the heaven itself by night, than the sun and the sun's light by day. Doubtless. Last of all, I imagine, he will be able to observe and [6]contemplate the natureof the sun, not as it appears in water or on alien ground, but as it is in itself inits own territory. Of course. His next step will be to draw the conclusion, that the sun is the author of theseasons and the years, and the guardian of all things in the visible world, and ina manner the cause of all those things which he and his companions used to see. Obviously, this will be his next step.... Now this imaginary case, my dear Glancon, you must apply in all its parts toour former statements, by comparing the region which the eye reveals, to theprison house, and the light of the fire therein to the power of the sun: and if, bythe upward ascent and the contemplation of the upper world, you understandthe mounting of the soul into the intellectual region, you will hit the tendencyof my own surmises, since you desire to be told what they are; though, indeed,God only knows whether they are correct. But, be that as it may, the viewwhich I take of the subject is to the following effect. In the world ofknowledge, the essential Form of Good is the limit of our enquiries, and canbarely be perceived; but, when perceived, we cannot help concluding that it isin every case the source of all that is bright and beautiful,—in the visible worldgiving birth to light and its master, and in the intellectual world dispensing,immediately and with full authority, truth and reason;—and that whosoeverwould act wisely, either in private or in public, must set this Form of Goodbefore his eyes." But in this passage, as throughout most of Plato's teaching, there is anidentification of the good with the truly real, which became embodied in thephilosophical[7]tradition, and is still largely operative in our own day. In thusallowing a legislative function to the good, Plato produced a divorce betweenphilosophy and science, from which, in my opinion, both have suffered eversince and are still suffering. The man of science, whatever his hopes may be,must lay them aside while he studies nature; and the philosopher, if he is toachieve truth must do the same. Ethical considerations can only legitimatelyappear when the truth has been ascertained: they can and should appear asdetermining our feeling towards the truth, and our manner of ordering our livesin view of the truth, but not as themselves dictating what the truth is to be. There are passages in Plato—among those which illustrate the scientific sideof his mind—where he seems clearly aware of this. The most noteworthy is theone in which Socrates, as a young man, is explaining the theory of ideas toParmenides. After Socrates has explained that there is an idea of the good, but not of suchthings as hair and mud and dirt, Parmenides advises him "not to despise eventhe meanest things," and this advice shows the genuine scientific temper. It iswith this impartial temper that the mystic's apparent insight into a higher realityand a hidden good has to be combined if philosophy is to realise its greatestpossibilities. And it is failure in this respect that has made so much of idealisticphilosophy thin, lifeless, and insubstantial. It is only in marriage with the worldthat our ideals can bear fruit: divorced from it, they remain barren. Butmarriage with the world is not to be achieved by an ideal which shrinks fromfact, or demands in advance that the world shall conform to its desires. Parmenides himself is the source of a peculiarly [8]interesting strain ofmysticism which pervades Plato's thought—the mysticism which may be called"logical" because it is embodied in theories on logic. This form of mysticism,which appears, so far as the West is concerned, to have originated withParmenides, dominates the reasonings of all the great mystical metaphysiciansfrom his day to that of Hegel and his modern disciples. Reality, he says, isuncreated, indestructible, unchanging, indivisible; it is "immovable in thebonds of mighty chains, without beginning and without end; since coming intobeing and passing away have been driven afar, and true belief has cast themaway." The fundamental principle of his inquiry is stated in a sentence whichwould not be out of place in Hegel: "Thou canst not know what is not—that isimpossible—nor utter it; for it is the same thing that can be thought and thatcan be." And again: "It needs must be that what can be thought and spoken ofis; for it is possible for it to be, and it is not possible for what is nothing to be."The impossibility of change follows from this principle; for what is past can bespoken of, and therefore, by the principle, still is. Mystical philosophy, in all ages and in all parts of the world, is characterisedby certain beliefs which are illustrated by the doctrines we have beenconsidering. There is, first, the belief in insight as against discursive analytic knowledge:the belief in a way of wisdom, sudden, penetrating, coercive, which iscontrasted with the slow and fallible study of outward appearance by a sciencerelying wholly upon the senses. All who are capable of absorption in an inwardpassion must have experienced at times the strange feeling of unreality incommon objects, the loss of contact with daily things, in which the solidity ofthe outer world is lost, and the soul[9]seems, in utter loneliness, to bring forth,out of its own depths, the mad dance of fantastic phantoms which have hithertoappeared as independently real and living. This is the negative side of themystic's initiation: the doubt concerning common knowledge, preparing theway for the reception of what seems a higher wisdom. Many men to whom thisnegative experience is familiar do not pass beyond it, but for the mystic it ismerely the gateway to an ampler world. The mystic insight begins with the sense of a mystery unveiled, of a hiddenwisdom now suddenly become certain beyond the possibility of a doubt. Thesense of certainty and revelation comes earlier than any definite belief. Thedefinite beliefs at which mystics arrive are the result of reflection upon theinarticulate experience gained in the moment of insight. Often, beliefs whichhave no real connection with this moment become subsequently attracted intothe central nucleus; thus in addition to the convictions which all mystics share,we find, in many of them, other convictions of a more local and temporarycharacter, which no doubt become amalgamated with what was essentiallymystical in virtue of their subjective certainty. We may ignore such inessentialaccretions, and confine ourselves to the beliefs which all mystics share. The first and most direct outcome of the moment of illumination is belief inthe possibility of a way of knowledge which may be called revelation or insightor intuition, as contrasted with sense, reason, and analysis, which are regardedas blind guides leading to the morass of illusion. Closely connected with thisbelief is the conception of a Reality behind the world of appearance and utterlydifferent from it. This Reality is regarded with an admiration often amountingto worship; it is [10]felt to be always and everywhere close at hand, thinly veiledby the shows of sense, ready, for the receptive mind, to shine in its glory eventhrough the apparent folly and wickedness of Man. The poet, the artist, and thelover are seekers after that glory: the haunting beauty that they pursue is thefaint reflection of its sun. But the mystic lives in the full light of the vision:what others dimly seek he knows, with a knowledge beside which all otherknowledge is ignorance. The second characteristic of mysticism is its belief in unity, and its refusal toadmit opposition or division anywhere. We found Heraclitus saying "good andill are one"; and again he says, "the way up and the way down is one and thesame." The same attitude appears in the simultaneous assertion of contradictorypropositions, such as: "We step and do not step into the same rivers; we are andare not." The assertion of Parmenides, that reality is one and indivisible, comesfrom the same impulse towards unity. In Plato, this impulse is less prominent,being held in check by his theory of ideas; but it reappears, so far as his logicpermits, in the doctrine of the primacy of the Good. A third mark of almost all mystical metaphysics is the denial of the reality ofTime. This is an outcome of the denial of division; if all is one, the distinctionof past and future must be illusory. We have seen this doctrine prominent inParmenides; and among moderns it is fundamental in the systems of Spinozaand Hegel. The last of the doctrines of mysticism which we have to consider is its beliefthat all evil is mere appearance, an illusion produced by the divisions andoppositions of the analytic intellect. Mysticism does not maintain that suchthings as cruelty, for example, are good, but it denies that they are real: theybelong to that lower [11]world of phantoms from which we are to be liberated bythe insight of the vision. Sometimes—for example in Hegel, and at leastverbally in Spinoza—not only evil, but good also, is regarded as illusory,though nevertheless the emotional attitude towards what is held to be Reality issuch as would naturally be associated with the belief that Reality is good. Whatis, in all cases, ethically characteristic of mysticism is absence of indignation orprotest, acceptance with joy, disbelief in the ultimate truth of the division intotwo hostile camps, the good and the bad. This attitude is a direct outcome of thenature of the mystical experience: with its sense of unity is associated a feelingof infinite peace. Indeed it may be suspected that the feeling of peace produces,as feelings do in dreams, the whole system of associated beliefs which make upthe body of mystic doctrine. But this is a difficult question, and one on which itcannot be hoped that mankind will reach agreement. Four questions thus arise in considering the truth or falsehood of mysticism,namely: I. Are there two ways of knowing, which may be called respectively reasonand intuition? And if so, is either to be preferred to the other? II. Is all plurality and division illusory? III. Is time unreal? IV. What kind of reality belongs to good and evil? On all four of these questions, while fully developed mysticism seems to memistaken, I yet believe that, by sufficient restraint, there is an element ofwisdom to be learned from the mystical way of feeling, which does not seem tobe attainable in any other manner. If this is the truth, mysticism is to becommended as an attitude towards life, not as a creed about the world.The [12]meta-physical creed, I shall maintain, is a mistaken outcome of theemotion, although this emotion, as colouring and informing all other thoughtsand feelings, is the inspirer of whatever is best in Man. Even the cautious andpatient investigation of truth by science, which seems the very antithesis of themystic's swift certainty, may be fostered and nourished by that very spirit ofreverence in which mysticism lives and moves.

I. REASON AND INTUITION[3]

Of the reality or unreality of the mystic's world I know nothing. I have no

wish to deny it, nor even to declare that the insight which reveals it is not agenuine insight. What I do wish to maintain—and it is here that the scientificattitude becomes imperative—is that insight, untested and unsupported, is aninsufficient guarantee of truth, in spite of the fact that much of the mostimportant truth is first suggested by its means. It is common to speak of anopposition between instinct and reason; in the eighteenth century, theopposition was drawn in favour of reason, but under the influence of Rousseauand the romantic movement instinct was given the preference, first by thosewho rebelled against artificial forms of government and thought, and then, asthe purely rationalistic defence of traditional theology became increasinglydifficult, by all who felt in science a menace to creeds which they associatedwith a spiritual outlook on life and the world. Bergson, under the name of"intuition," has raised instinct to the position of sole [13]arbiter of metaphysicaltruth. But in fact the opposition of instinct and reason is mainly illusory.Instinct, intuition, or insight is what first leads to the beliefs which subsequentreason confirms or confutes; but the confirmation, where it is possible, consists,in the last analysis, of agreement with other beliefs no less instinctive. Reasonis a harmonising, controlling force rather than a creative one. Even in the mostpurely logical realm, it is insight that first arrives at what is new. Where instinct and reason do sometimes conflict is in regard to single beliefs,held instinctively, and held with such determination that no degree ofinconsistency with other beliefs leads to their abandonment. Instinct, like allhuman faculties, is liable to error. Those in whom reason is weak are oftenunwilling to admit this as regards themselves, though all admit it in regard toothers. Where instinct is least liable to error is in practical matters as to whichright judgment is a help to survival: friendship and hostility in others, forinstance, are often felt with extraordinary discrimination through very carefuldisguises. But even in such matters a wrong impression may be given byreserve or flattery; and in matters less directly practical, such as philosophydeals with, very strong instinctive beliefs are sometimes wholly mistaken, aswe may come to know through their perceived inconsistency with other equallystrong beliefs. It is such considerations that necessitate the harmonisingmediation of reason, which tests our beliefs by their mutual compatibility, andexamines, in doubtful cases, the possible sources of error on the one side andon the other. In this there is no opposition to instinct as a whole, but only toblind reliance upon some one interesting aspect of instinct to the exclusion ofother more [14]commonplace but not less trustworthy aspects. It is such one-sidedness, not instinct itself, that reason aims at correcting. These more or less trite maxims may be illustrated by application toBergson's advocacy of "intuition" as against "intellect." There are, he says,"two profoundly different ways of knowing a thing. The first implies that wemove round the object: the second that we enter into it. The first depends on thepoint of view at which we are placed and on the symbols by which we expressourselves. The second neither depends on a point of view nor relies on anysymbol. The first kind of knowledge may be said to stop at the relative; thesecond, in those cases where it is possible, to attain the absolute."[4] Thesecond of these, which is intuition, is, he says, "the kind of intellectualsympathy by which one places oneself within an object in order to coincidewith what is unique in it and therefore inexpressible" (p. 6). In illustration, hementions self-knowledge: "there is one reality, at least, which we all seize fromwithin, by intuition and not by simple analysis. It is our own personality in itsflowing through time—our self which endures" (p. 8). The rest of Bergson'sphilosophy consists in reporting, through the imperfect medium of words, theknowledge gained by intuition, and the consequent complete condemnation ofall the pretended knowledge derived from science and common sense. This procedure, since it takes sides in a conflict of instinctive beliefs, standsin need of justification by proving the greater trustworthiness of the beliefs onone side than of those on the other. Bergson attempts this justification in twoways, first by explaining that intellect is a purely practical faculty to securebiological success, [15]secondly by mentioning remarkable feats of instinct inanimals and by pointing out characteristics of the world which, though intuitioncan apprehend them, are baffling to intellect as he interprets it. Of Bergson's theory that intellect is a purely practical faculty, developed inthe struggle for survival, and not a source of true beliefs, we may say, first, thatit is only through intellect that we know of the struggle for survival and of thebiological ancestry of man: if the intellect is misleading, the whole of thismerely inferred history is presumably untrue. If, on the other hand, we agreewith him in thinking that evolution took place as Darwin believed, then it is notonly intellect, but all our faculties, that have been developed under the stress ofpractical utility. Intuition is seen at its best where it is directly useful, forexample in regard to other people's characters and dispositions. Bergsonapparently holds that capacity, for this kind of knowledge is less explicable bythe struggle for existence than, for example, capacity for pure mathematics. Yetthe savage deceived by false friendship is likely to pay for his mistake with hislife; whereas even in the most civilised societies men are not put to death formathematical incompetence. All the most striking of his instances of intuitionin animals have a very direct survival value. The fact is, of course, that bothintuition and intellect have been developed because they are useful, and that,speaking broadly, they are useful when they give truth and become harmfulwhen they give falsehood. Intellect, in civilised man, like artistic capacity, hasoccasionally been developed beyond the point where it is useful to theindividual; intuition, on the other hand, seems on the whole to diminish ascivilisation increases. It is greater, as a rule, in children than in adults, in theuneducated than in the educated. [16]Probably in dogs it exceeds anything to befound in human beings. But those who see in these facts a recommendation ofintuition ought to return to running wild in the woods, dyeing themselves withwoad and living on hips and haws. Let us next examine whether intuition possesses any such infallibility asBergson claims for it. The best instance of it, according to him, is ouracquaintance with ourselves; yet self-knowledge is proverbially rare anddifficult. Most men, for example, have in their nature meannesses, vanities, andenvies of which they are quite unconscious, though even their best friends canperceive them without any difficulty. It is true that intuition has aconvincingness which is lacking to intellect: while it is present, it is almostimpossible to doubt its truth. But if it should appear, on examination, to be atleast as fallible as intellect, its greater subjective certainty becomes a demerit,making it only the more irresistibly deceptive. Apart from self-knowledge, oneof the most notable examples of intuition is the knowledge people believethemselves to possess of those with whom they are in love: the wall betweendifferent personalities seems to become transparent, and people think they seeinto another soul as into their own. Yet deception in such cases is constantlypractised with success; and even where there is no intentional deception,experience gradually proves, as a rule, that the supposed insight was illusory,and that the slower more groping methods of the intellect are in the long runmore reliable. Bergson maintains that intellect can only deal with things in so far as theyresemble what has been experienced in the past, while intuition has the powerof apprehending the uniqueness and novelty that always belong to each freshmoment. That there is something unique [17]and new at every moment, iscertainly true; it is also true that this cannot be fully expressed by means ofintellectual concepts. Only direct acquaintance can give knowledge of what isunique and new. But direct acquaintance of this kind is given fully in sensation,and does not require, so far as I can see, any special faculty of intuition for itsapprehension. It is neither intellect nor intuition, but sensation, that suppliesnew data; but when the data are new in any remarkable manner, intellect ismuch more capable of dealing with them than intuition would be. The hen witha brood of ducklings no doubt has intuition which seems to place her insidethem, and not merely to know them analytically; but when the ducklings take tothe water, the whole apparent intuition is seen to be illusory, and the hen is lefthelpless on the shore. Intuition, in fact, is an aspect and development ofinstinct, and, like all instinct, is admirable in those customary surroundingswhich have moulded the habits of the animal in question, but totallyincompetent as soon as the surroundings are changed in a way which demandssome non-habitual mode of action. The theoretical understanding of the world, which is the aim of philosophy, isnot a matter of great practical importance to animals, or to savages, or even tomost civilised men. It is hardly to be supposed, therefore, that the rapid, roughand ready methods of instinct or intuition will find in this field a favourableground for their application. It is the older kinds of activity, which bring out ourkinship with remote generations of animal and semi-human ancestors, thatshow intuition at its best. In such matters as self-preservation and love, intuitionwill act sometimes (though not always) with a swiftness and precision whichare astonishing to the critical intellect. But philosophy is not one ofthe [18]pursuits which illustrate our affinity with the past: it is a highly refined,highly civilised pursuit, demanding, for its success, a certain liberation from thelife of instinct, and even, at times, a certain aloofness from all mundane hopesand fears. It is not in philosophy, therefore, that we can hope to see intuition atits best. On the contrary, since the true objects of philosophy, and the habit ofthought demanded for their apprehension, are strange, unusual, and remote, it ishere, more almost than anywhere else, that intellect proves superior to intuition,and that quick unanalysed convictions are least deserving of uncriticalacceptance. In advocating the scientific restraint and balance, as against the self-assertionof a confident reliance upon intuition, we are only urging, in the sphere ofknowledge, that largeness of contemplation, that impersonal disinterestedness,and that freedom from practical preoccupations which have been inculcated byall the great religions of the world. Thus our conclusion, however it mayconflict with the explicit beliefs of many mystics, is, in essence, not contrary tothe spirit which inspires those beliefs, but rather the outcome of this very spiritas applied in the realm of thought.

II. UNITY AND PLURALITY

One of the most convincing aspects of the mystic illumination is the apparentrevelation of the oneness of all things, giving rise to pantheism in religion andto monism in philosophy. An elaborate logic, beginning with Parmenides, andculminating in Hegel and his followers, has been gradually developed, to provethat the universe is one indivisible Whole, and that what seem to be its parts, ifconsidered as substantial and [19]self-existing, are mere illusion. The conceptionof a Reality quite other than the world of appearance, a reality one, indivisible,and unchanging, was introduced into Western philosophy by Parmenides, not,nominally at least, for mystical or religious reasons, but on the basis of a logicalargument as to the impossibility of not-being, and most subsequentmetaphysical systems are the outcome of this fundamental idea. The logic used in defence of mysticism seems to be faulty as logic, and opento technical criticisms, which I have explained elsewhere. I shall not here repeatthese criticisms, since they are lengthy and difficult, but shall instead attemptan analysis of the state of mind from which mystical logic has arisen. Belief in a reality quite different from what appears to the senses arises withirresistible force in certain moods, which are the source of most mysticism, andof most metaphysics. While such a mood is dominant, the need of logic is notfelt, and accordingly the more thoroughgoing mystics do not employ logic, butappeal directly to the immediate deliverance of their insight. But such fullydeveloped mysticism is rare in the West. When the intensity of emotionalconviction subsides, a man who is in the habit of reasoning will search forlogical grounds in favour of the belief which he finds in himself. But since thebelief already exists, he will be very hospitable to any ground that suggestsitself. The paradoxes apparently proved by his logic are really the paradoxes ofmysticism, and are the goal which he feels his logic must reach if it is to be inaccordance with insight. The resulting logic has rendered most philosophersincapable of giving any account of the world of science and daily life. If theyhad been anxious to give such an account, they would probably havediscovered the errors of their [20]logic; but most of them were less anxious tounderstand the world of science and daily life than to convict it of unreality inthe interests of a super-sensible "real" world. It is in this way that logic has been pursued by those of the great philosopherswho were mystics. But since they usually took for granted the supposed insightof the mystic emotion, their logical doctrines were presented with a certaindryness, and were believed by their disciples to be quite independent of thesudden illumination from which they sprang. Nevertheless their origin clung tothem, and they remained—to borrow a useful word from Mr. Santayana—"malicious" in regard to the world of science and common sense. It is only sothat we can account for the complacency with which philosophers haveaccepted the inconsistency of their doctrines with all the common and scientificfacts which seem best established and most worthy of belief. The logic of mysticism shows, as is natural, the defects which are inherent inanything malicious. The impulse to logic, not felt while the mystic mood isdominant, reasserts itself as the mood fades, but with a desire to retain thevanishing insight, or at least to prove that it was insight, and that what seems tocontradict it is illusion. The logic which thus arises is not quite disinterested orcandid, and is inspired by a certain hatred of the daily world to which it is to beapplied. Such an attitude naturally does not tend to the best results. Everyoneknows that to read an author simply in order to refute him is not the way tounderstand him; and to read the book of Nature with a conviction that it is allillusion is just as unlikely to lead to understanding. If our logic is to find thecommon world intelligible, it must not be hostile, but must be inspired by agenuine [21]acceptance such as is not usually to be found amongmetaphysicians.

III. TIME

The unreality of time is a cardinal doctrine of many metaphysical systems,

often nominally based, as already by Parmenides, upon logical arguments, butoriginally derived, at any rate in the founders of new systems, from thecertainty which is born in the moment of mystic insight. As a Persian Sufi poetsays:"Past and future are what veil God from our sight.Burn up both of them withfire! How longWilt thou be partitioned by these segments as a reed?"[5] The belief that what is ultimately real must be immutable is a very commonone: it gave rise to the metaphysical notion of substance, and finds, even now, awholly illegitimate satisfaction in such scientific doctrines as the conservationof energy and mass. It is difficult to disentangle the truth and the error in this view. Thearguments for the contention that time is unreal and that the world of sense isillusory must, I think, be regarded as fallacious. Nevertheless there is somesense—easier to feel than to state—in which time is an unimportant andsuperficial characteristic of reality. Past and future must be acknowledged to beas real as the present, and a certain emancipation from slavery to time isessential to philosophic thought. The importance of time is rather practical thantheoretical, rather in relation to our desires than in relation to truth. A truerimage of the world, I think, is obtained by picturing things as entering into thestream of time from an eternal world outside, than from a view which regardstime as the devouring tyrant of all that is. Both in [22]thought and in feeling,even though time be real, to realise the unimportance of time is the gate ofwisdom. That this is the case may be seen at once by asking ourselves why ourfeelings towards the past are so different from our feelings towards the future.The reason for this difference is wholly practical: our wishes can affect thefuture but not the past, the future is to some extent subject to our power, whilethe past is unalterably fixed. But every future will some day be past: if we seethe past truly now, it must, when it was still future, have been just what we nowsee it to be, and what is now future must be just what we shall see it to be whenit has become past. The felt difference of quality between past and future,therefore, is not an intrinsic difference, but only a difference in relation to us: toimpartial contemplation, it ceases to exist. And impartiality of contemplation is,in the intellectual sphere, that very same virtue of disinterestedness which, inthe sphere of action, appears as justice and unselfishness. Whoever wishes tosee the world truly, to rise in thought above the tyranny of practical desires,must learn to overcome the difference of attitude towards past and future, andto survey the whole stream of time in one comprehensive vision. The kind of way in which, as it seems to me, time ought not to enter into ourtheoretic philosophical thought, may be illustrated by the philosophy which hasbecome associated with the idea of evolution, and which is exemplified byNietzsche, pragmatism, and Bergson. This philosophy, on the basis of thedevelopment which has led from the lowest forms of life up to man, seesin progress the fundamental law of the universe, and thus admits the differencebetween earlier and later into the very citadel of its contemplative outlook.With its past and future [23]history of the world, conjectural as it is, I do notwish to quarrel. But I think that, in the intoxication of a quick success, muchthat is required for a true understanding of the universe has been forgotten.Something of Hellenism, something, too, of Oriental resignation, must becombined with its hurrying Western self-assertion before it can emerge fromthe ardour of youth into the mature wisdom of manhood. In spite of its appealsto science, the true scientific philosophy, I think, is something more arduousand more aloof, appealing to less mundane hopes, and requiring a severerdiscipline for its successful practice. Darwin's Origin of Species persuaded the world that the difference betweendifferent species of animals and plants is not the fixed immutable differencethat it appears to be. The doctrine of natural kinds, which had renderedclassification easy and definite, which was enshrined in the Aristoteliantradition, and protected by its supposed necessity for orthodox dogma, wassuddenly swept away for ever out of the biological world. The differencebetween man and the lower animals, which to our human conceit appearsenormous, was shown to be a gradual achievement, involving intermediatebeing who could not with certainty be placed either within or without thehuman family. The sun and the planets had already been shown by Laplace tobe very probably derived from a primitive more or less undifferentiated nebula.Thus the old fixed landmarks became wavering and indistinct, and all sharpoutlines were blurred. Things and species lost their boundaries, and none couldsay where they began or where they ended. But if human conceit was staggered for a moment by its kinship with the ape,it soon found a way to reassert itself, and that way is the "philosophy" ofevolution. [24]A process which led from the am[oe]ba to Man appeared to thephilosophers to be obviously a progress—though whether the am[oe]ba wouldagree with this opinion is not known. Hence the cycle of changes which sciencehad shown to be the probable history of the past was welcomed as revealing alaw of development towards good in the universe—an evolution or unfoldingof an idea slowly embodying itself in the actual. But such a view, though itmight satisfy Spencer and those whom we may call Hegelian evolutionists,could not be accepted as adequate by the more whole-hearted votaries ofchange. An ideal to which the world continuously approaches is, to theseminds, too dead and static to be inspiring. Not only the aspiration, but the idealtoo, must change and develop with the course of evolution: there must be nofixed goal, but a continual fashioning of fresh needs by the impulse which islife and which alone gives unity to the process. Life, in this philosophy, is a continuous stream, in which all divisions areartificial and unreal. Separate things, beginnings and endings, are mereconvenient fictions: there is only smooth unbroken transition. The beliefs of to-day may count as true to-day, if they carry us along the stream; but to-morrowthey will be false, and must be replaced by new beliefs to meet the newsituation. All our thinking consists of convenient fictions, imaginarycongealings of the stream: reality flows on in spite of all our fictions, andthough it can be lived, it cannot be conceived in thought. Somehow, withoutexplicit statement, the assurance is slipped in that the future, though we cannotforesee it, will be better than the past or the present: the reader is like the childwhich expects a sweet because it has been told to open its mouth and shut itseyes. Logic, mathematics, [25]physics disappear in this philosophy, because theyare too "static"; what is real is no impulse and movement towards a goal which,like the rainbow, recedes as we advance, and makes every place different whenit reaches it from what it appeared to be at a distance. I do not propose to enter upon a technical examination of this philosophy. Iwish only to maintain that the motives and interests which inspire it are soexclusively practical, and the problems with which it deals are so special, that itcan hardly be regarded as touching any of the questions that, to my mind,constitute genuine philosophy. The predominant interest of evolutionism is in the question of human destiny,or at least of the destiny of Life. It is more interested in morality and happinessthan in knowledge for its own sake. It must be admitted that the same may besaid of many other philosophies, and that a desire for the kind of knowledgewhich philosophy can give is very rare. But if philosophy is to attain truth, it isnecessary first and foremost that philosophers should acquire the disinterestedintellectual curiosity which characterises the genuine man of science.Knowledge concerning the future—which is the kind of knowledge that mustbe sought if we are to know about human destiny—is possible within certainnarrow limits. It is impossible to say how much the limits may be enlarged withthe progress of science. But what is evident is that any proposition about thefuture belongs by its subject-matter to some particular science, and is to beascertained, if at all, by the methods of that science. Philosophy is not a shortcut to the same kind of results as those of the other sciences: if it is to be agenuine study, it must have a province of its own, and aim at results which theother sciences can neither prove nor disprove. [26]Evolutionism, in basing itself upon the notion of progress, which ischange from the worse to the better, allows the notion of time, as it seems tome, to become its tyrant rather than its servant, and thereby loses thatimpartiality of contemplation which is the source of all that is best inphilosophic thought and feeling. Metaphysicians, as we saw, have frequentlydenied altogether the reality of time. I do not wish to do this; I wish only topreserve the mental outlook which inspired the denial, the attitude which, inthought, regards the past as having the same reality as the present and the sameimportance as the future. "In so far," says Spinoza,[6] "as the mind conceives athing according to the dictate of reason, it will be equally affected whether theidea is that of a future, past, or present thing." It is this "conceiving accordingto the dictate of reason" that I find lacking in the philosophy which is based onevolution.

IV. GOOD AND EVIL

Mysticism maintains that all evil is illusory, and sometimes maintains thesame view as regards good, but more often holds that all Reality is good. Bothviews are to be found in Heraclitus: "Good and ill are one," he says, but again,"To God all things are fair and good and right, but men hold some things wrongand some right." A similar twofold position is to be found in Spinoza, but heuses the word "perfection" when he means to speak of the good that is notmerely human. "By reality and perfection I mean the same thing," hesays;[7] but elsewhere we find the definition: "By good I shall mean that whichwe certainly know to be useful to us."[8] Thus perfection belongs to Reality inits own nature, but [27]goodness is relative to ourselves and our needs, anddisappears in an impartial survey. Some such distinction, I think, is necessaryin order to understand the ethical outlook of mysticism: there is a lowermundane kind of good and evil, which divides the world of appearance intowhat seem to be conflicting parts; but there is also a higher, mystical kind ofgood, which belongs to Reality and is not opposed by any correlative kind ofevil. It is difficult to give a logically tenable account of this position withoutrecognising that good and evil are subjective, that what is good is merely thattowards which we have one kind of feeling, and what is evil is merely thattowards which we have another kind of feeling. In our active life, where wehave to exercise choice, and to prefer this to that of two possible acts, it isnecessary to have a distinction of good and evil, or at least of better and worse.But this distinction, like everything pertaining to action, belongs to whatmysticism regards as the world of illusion, if only because it is essentiallyconcerned with time. In our contemplative life, where action is not called for, itis possible to be impartial, and to overcome the ethical dualism which actionrequires. So long as we remain merely impartial, we may be content to say thatboth the good and the evil of action are illusions. But if, as we must do if wehave the mystic vision, we find the whole world worthy of love and worship, ifwe see"The earth, and every common sight....Apparell'd in celestial light,"we shall say that there is a higher good than that of action, and that this highergood belongs to the whole world as it is in reality. In this way the twofoldattitude and the apparent vacillation of mysticism are explained and justified. [28]The possibility of this universal love and joy in all that exists is ofsupreme importance for the conduct and happiness of life, and givesinestimable value to the mystic emotion, apart from any creeds which may bebuilt upon it. But if we are not to be led into false beliefs, it is necessary torealise exactly what the mystic emotion reveals. It reveals a possibility ofhuman nature—a possibility of a nobler, happier, freer life than any that can beotherwise achieved. But it does not reveal anything about the non-human, orabout the nature of the universe in general. Good and bad, and even the highergood that mysticism finds everywhere, are the reflections of our own emotionson other things, not part of the substance of things as they are in themselves.And therefore an impartial contemplation, freed from all pre-occupation withSelf, will not judge things good or bad, although it is very easily combined withthat feeling of universal love which leads the mystic to say that the whole worldis good. The philosophy of evolution, through the notion of progress, is bound upwith the ethical dualism of the worse and the better, and is thus shut out, notonly from the kind of survey which discards good and evil altogether from itsview, but also from the mystical belief in the goodness of everything. In thisway the distinction of good and evil, like time, becomes a tyrant in thisphilosophy, and introduces into thought the restless selectiveness of action.Good and evil, like time, are, it would seem, not general or fundamental in theworld of thought, but late and highly specialised members of the intellectualhierarchy. Although, as we saw, mysticism can be interpreted so as to agree with theview that good and evil are not intellectually fundamental, it must be admittedthat here [29]we are no longer in verbal agreement with most of the greatphilosophers and religious teachers of the past. I believe, however, that theelimination of ethical considerations from philosophy is both scientificallynecessary and—though this may seem a paradox—an ethical advance. Boththese contentions must be briefly defended. The hope of satisfaction to our more human desires—the hope ofdemonstrating that the world has this or that desirable ethical characteristic—isnot one which, so far as I can see, a scientific philosophy can do anythingwhatever to satisfy. The difference between a good world and a bad one is adifference in the particular characteristics of the particular things that exist inthese worlds: it is not a sufficiently abstract difference to come within theprovince of philosophy. Love and hate, for example, are ethical opposites, butto philosophy they are closely analogous attitudes towards objects. The generalform and structure of those attitudes towards objects which constitute mentalphenomena is a problem for philosophy, but the difference between love andhate is not a difference of form or structure, and therefore belongs rather to thespecial science of psychology than to philosophy. Thus the ethical interestswhich have often inspired philosophers must remain in the background: somekind of ethical interest may inspire the whole study, but none must obtrude inthe detail or be expected in the special results which are sought. If this view seems at first sight disappointing, we may remind ourselves thata similar change has been found necessary in all the other sciences. Thephysicist or chemist is not now required to prove the ethical importance of hisions or atoms; the biologist is not expected to prove the utility of the plants oranimals [30]which he dissects. In pre-scientific ages this was not the case.Astronomy, for example, was studied because men believed in astrology: it wasthought that the movements of the planets had the most direct and importantbearing upon the lives of human beings. Presumably, when this belief decayedand the disinterested study of astronomy began, many who had found astrologyabsorbingly interesting decided that astronomy had too little human interest tobe worthy of study. Physics, as it appears in Plato's Timæus for example, is fullof ethical notions: it is an essential part of its purpose to show that the earth isworthy of admiration. The modern physicist, on the contrary, though he has nowish to deny that the earth is admirable, is not concerned, as physicist, with itsethical attributes: he is merely concerned to find out facts, not to considerwhether they are good or bad. In psychology, the scientific attitude is evenmore recent and more difficult than in the physical sciences: it is natural toconsider that human nature is either good or bad, and to suppose that thedifference between good and bad, so all-important in practice, must beimportant in theory also. It is only during the last century that an ethicallyneutral psychology has grown up; and here too, ethical neutrality has beenessential to scientific success. In philosophy, hitherto, ethical neutrality has been seldom sought and hardlyever achieved. Men have remembered their wishes, and have judgedphilosophies in relation to their wishes. Driven from the particular sciences, thebelief that the notions of good and evil must afford a key to the understandingof the world has sought a refuge in philosophy. But even from this last refuge,if philosophy is not to remain a set of pleasing dreams, this belief must bedriven forth. It is a commonplace that[31]happiness is not best achieved by thosewho seek it directly; and it would seem that the same is true of the good. Inthought, at any rate, those who forget good and evil and seek only to know thefacts are more likely to achieve good than those who view the world throughthe distorting medium of their own desires. We are thus brought back to our seeming paradox, that a philosophy whichdoes not seek to impose upon the world its own conceptions of good and evil isnot only more likely to achieve truth, but is also the outcome of a higher ethicalstandpoint than one which, like evolutionism and most traditional systems, isperpetually appraising the universe and seeking to find in it an embodiment ofpresent ideals. In religion, and in every deeply serious view of the world and ofhuman destiny, there is an element of submission, a realisation of the limits ofhuman power, which is somewhat lacking in the modern world, with its quickmaterial successes and its insolent belief in the boundless possibilities ofprogress. "He that loveth his life shall lose it"; and there is danger lest, througha too confident love of life, life itself should lose much of what gives it itshighest worth. The submission which religion inculcates in action is essentiallythe same in spirit as that which science teaches in thought; and the ethicalneutrality by which its victories have been achieved is the outcome of thatsubmission. The good which it concerns us to remember is the good which it lies in ourpower to create—the good in our own lives and in our attitude towards theworld. Insistence on belief in an external realisation of the good is a form ofself-assertion, which, while it cannot secure the external good which it desires,can seriously impair the inward good which lies within our power, and destroythat reverence towards fact which constitutes both what is [32]valuable inhumility and what is fruitful in the scientific temper. Human beings cannot, of course, wholly transcend human nature; somethingsubjective, if only the interest that determines the direction of our attention,must remain in all our thought. But scientific philosophy comes nearer toobjectivity than any other human pursuit, and gives us, therefore, the closestconstant and the most intimate relation with the outer world that it is possible toachieve. To the primitive mind, everything is either friendly or hostile; butexperience has shown that friendliness and hostility are not the conceptions bywhich the world is to be understood. Scientific philosophy thus represents,though as yet only in a nascent condition, a higher form of thought than anypre-scientific belief or imagination, and, like every approach to self-transcendence, it brings with it a rich reward in increase of scope and breadthand comprehension. Evolutionism, in spite of its appeals to particular scientificfacts, fails to be a truly scientific philosophy because of its slavery to time, itsethical preoccupations, and its predominant interest in our mundane concernsand destiny. A truly scientific philosophy will be more humble, morepiecemeal, more arduous, offering less glitter of outward mirage to flatterfallacious hopes, but more indifferent to fate, and more capable of accepting theworld without the tyrannous imposition of our human and temporary demands.

FOOTNOTES:

[1]All the above quotations are from Burnet's Early Greek Philosophy, (2nd ed., 1908), pp. 146-156.[2]Republic, 514, translated by Davies and Vaughan.[3]This section, and also one or two pages in later sections, have been printed in a course of Lowelllectures On our knowledge of the external world, published by the Open Court Publishing Company.But I have left them here, as this is the context for which they were originally written.[4]Introduction to Metaphysics, p. 1.[5]Whinfield's translation of the Masnavi (Trübner, 1887), p. 34.[6]Ethics, Bk. IV, Prop. LXII.[7]Ib., Pt. IV, Df. I.[8]Ethics. Pt. II. Df. VI.

[33]

IIToC

THE PLACE OF SCIENCE IN A LIBERAL EDUCATION

Science, to the ordinary reader of newspapers, is represented by a varying

selection of sensational triumphs, such as wireless telegraphy and aeroplanesradio-activity and the marvels of modern alchemy. It is not of this aspect ofscience that I wish to speak. Science, in this aspect, consists of detached up-to-date fragments, interesting only until they are replaced by something newer andmore up-to-date, displaying nothing of the systems of patiently constructedknowledge out of which, almost as a casual incident, have come the practicallyuseful results which interest the man in the street. The increased command overthe forces of nature which is derived from science is undoubtedly an amplysufficient reason for encouraging scientific research, but this reason has been sooften urged and is so easily appreciated that other reasons, to my mind quite asimportant, are apt to be overlooked. It is with these other reasons, especiallywith the intrinsic value of a scientific habit of mind in forming our outlook onthe world, that I shall be concerned in what follows. The instance of wireless telegraphy will serve to illustrate the differencebetween the two points of view. Almost all the serious intellectual labourrequired for the[34]possibility of this invention is due to three men—Faraday,Maxwell, and Hertz. In alternating layers of experiment and theory these threemen built up the modern theory of electromagnetism, and demonstrated theidentity of light with electromagnetic waves. The system which they discoveredis one of profound intellectual interest, bringing together and unifying anendless variety of apparently detached phenomena, and displaying a cumulativemental power which cannot but afford delight to every generous spirit. Themechanical details which remained to be adjusted in order to utilise theirdiscoveries for a practical system of telegraphy demanded, no doubt, veryconsiderable ingenuity, but had not that broad sweep and that universalitywhich could give them intrinsic interest as an object of disinterestedcontemplation. From the point of view of training the mind, of giving that well-informed,impersonal outlook which constitutes culture in the good sense of this much-misused word, it seems to be generally held indisputable that a literaryeducation is superior to one based on science. Even the warmest advocates ofscience are apt to rest their claims on the contention that culture ought to besacrificed to utility. Those men of science who respect culture, when theyassociate with men learned in the classics, are apt to admit, not merely politely,but sincerely, a certain inferiority on their side, compensated doubtless by theservices which science renders to humanity, but none the less real. And so longas this attitude exists among men of science, it tends to verify itself: theintrinsically valuable aspects of science tend to be sacrificed to the merelyuseful, and little attempt is made to preserve that leisurely, systematic surveyby which the finer quality of mind is formed and nourished. [35]But even if there be, in present fact, any such inferiority as is supposed inthe educational value of science, this is, I believe, not the fault of science itself,but the fault of the spirit in which science is taught. If its full possibilities wererealised by those who teach it, I believe that its capacity of producing thosehabits of mind which constitute the highest mental excellence would be at leastas great as that of literature, and more particularly of Greek and Latin literature.In saying this I have no wish whatever to disparage a classical education. I havenot myself enjoyed its benefits, and my knowledge of Greek and Latin authorsis derived almost wholly from translations. But I am firmly persuaded that theGreeks fully deserve all the admiration that is bestowed upon them, and that itis a very great and serious loss to be unacquainted with their writings. It is notby attacking them, but by drawing attention to neglected excellences in science,that I wish to conduct my argument. One defect, however, does seem inherent in a purely classical education—namely, a too exclusive emphasis on the past. By the study of what isabsolutely ended and can never be renewed, a habit of criticism towards thepresent and the future is engendered. The qualities in which the present excelsare qualities to which the study of the past does not direct attention, and towhich, therefore, the student of Greek civilisation may easily become blind. Inwhat is new and growing there is apt to be something crude, insolent, even alittle vulgar, which is shocking to the man of sensitive taste; quivering from therough contact, he retires to the trim gardens of a polished past, forgetting thatthey were reclaimed from the wilderness by men as rough and earth-soiled asthose from whom he shrinks in his own day. The habit of being unable torecognise merit [36]until it is dead is too apt to be the result of a purely bookishlife, and a culture based wholly on the past will seldom be able to piercethrough everyday surroundings to the essential splendour of contemporarythings, or to the hope of still greater splendour in the future."My eyes saw not the men of old;And now their age away has rolled.I weep—to think I shall not seeThe heroes of posterity."So says the Chinese poet; but such impartiality is rare in the more pugnaciousatmosphere of the West, where the champions of past and future fight a never-ending battle, instead of combining to seek out the merits of both. This consideration, which militates not only against the exclusive study ofthe classics, but against every form of culture which has become static,traditional, and academic, leads inevitably to the fundamental question: What isthe true end of education? But before attempting to answer this question it willbe well to define the sense in which we are to use the word "education." Forthis purpose I shall distinguish the sense in which I mean to use it from twoothers, both perfectly legitimate, the one broader and the other narrower thanthe sense in which I mean to use the word. In the broader sense, education will include not only what we learn throughinstruction, but all that we learn through personal experience—the formation ofcharacter through the education of life. Of this aspect of education, vitallyimportant as it is, I will say nothing, since its consideration would introducetopics quite foreign to the question with which we are concerned. In the narrower sense, education may be confined to instruction, theimparting of definite information on [37]various subjects, because suchinformation, in and for itself, is useful in daily life. Elementary education—reading, writing, and arithmetic—is almost wholly of this kind. But instruction,necessary as it is, does not per se constitute education in the sense in which Iwish to consider it. Education, in the sense in which I mean it, may be defined as the formation,by means of instruction, of certain mental habits and a certain outlook on lifeand the world. It remains to ask ourselves, what mental habits, and what sort ofoutlook, can be hoped for as the result of instruction? When we have answeredthis question we can attempt to decide what science has to contribute to theformation of the habits and outlook which we desire. Our whole life is built about a certain number—not a very small number—ofprimary instincts and impulses. Only what is in some way connected with theseinstincts and impulses appears to us desirable or important; there is no faculty,whether "reason" or "virtue" or whatever it may be called, that can take ouractive life and our hopes and fears outside the region controlled by these firstmovers of all desire. Each of them is like a queen-bee, aided by a hive ofworkers gathering honey; but when the queen is gone the workers languish anddie, and the cells remain empty of their expected sweetness. So with eachprimary impulse in civilised man: it is surrounded and protected by a busyswarm of attendant derivative desires, which store up in its service whateverhoney the surrounding world affords. But if the queen-impulse dies, the death-dealing influence, though retarded a little by habit, spreads slowly through allthe subsidiary impulses, and a whole tract of life becomes inexplicablycolourless. What was formerly full of zest, and so obviously worth doing that itraised [38]no questions, has now grown dreary and purposeless: with a sense ofdisillusion we inquire the meaning of life, and decide, perhaps, that all isvanity. The search for an outside meaning that can compel an inner responsemust always be disappointed: all "meaning" must be at bottom related to ourprimary desires, and when they are extinct no miracle can restore to the worldthe value which they reflected upon it. The purpose of education, therefore, cannot be to create any primary impulsewhich is lacking in the uneducated; the purpose can only be to enlarge thescope of those that human nature provides, by increasing the number andvariety of attendant thoughts, and by showing where the most permanentsatisfaction is to be found. Under the impulse of a Calvinistic horror of the"natural man," this obvious truth has been too often misconceived in thetraining of the young; "nature" has been falsely regarded as excluding all that isbest in what is natural, and the endeavour to teach virtue has led to theproduction of stunted and contorted hypocrites instead of full-grown humanbeings. From such mistakes in education a better psychology or a kinder heartis beginning to preserve the present generation; we need, therefore, waste nomore words on the theory that the purpose of education is to thwart or eradicatenature. But although nature must supply the initial force of desire, nature is not, inthe civilised man, the spasmodic, fragmentary, and yet violent set of impulsesthat it is in the savage. Each impulse has its constitutional ministry of thoughtand knowledge and reflection, through which possible conflicts of impulses areforeseen, and temporary impulses are controlled by the unifying impulse whichmay be called wisdom. In this way [39]education destroys the crudity of instinct,and increases through knowledge the wealth and variety of the individual'scontacts with the outside world, making him no longer an isolated fighting unit,but a citizen of the universe, embracing distant countries, remote regions ofspace, and vast stretches of past and future within the circle of his interests. It isthis simultaneous softening in the insistence of desire and enlargement of itsscope that is the chief moral end of education. Closely connected with this moral end is the more purely intellectual aim ofeducation, the endeavour to make us see and imagine the world in an objectivemanner, as far as possible as it is in itself, and not merely through the distortingmedium of personal desire. The complete attainment of such an objective viewis no doubt an ideal, indefinitely approachable, but not actually and fullyrealisable. Education, considered as a process of forming our mental habits andour outlook on the world, is to be judged successful in proportion as itsoutcome approximates to this ideal; in proportion, that is to say, as it gives us atrue view of our place in society, of the relation of the whole human society toits non-human environment, and of the nature of the non-human world as it isin itself apart from our desires and interests. If this standard is admitted, we canreturn to the consideration of science, inquiring how far science contributes tosuch an aim, and whether it is in any respect superior to its rivals in educationalpractice.

II

Two opposite and at first sight conflicting merits belong to science as againstliterature and art. The one, which is not inherently necessary, but is certainlytrue [40]at the present day, is hopefulness as to the future of human achievement,and in particular as to the useful work that may be accomplished by anyintelligent student. This merit and the cheerful outlook which it engendersprevent what might otherwise be the depressing effect of another aspect ofscience, to my mind also a merit, and perhaps its greatest merit—I mean theirrelevance of human passions and of the whole subjective apparatus wherescientific truth is concerned. Each of these reasons for preferring the study ofscience requires some amplification. Let us begin with the first. In the study of literature or art our attention is perpetually riveted upon thepast: the men of Greece or of the Renaissance did better than any men do now;the triumphs of former ages, so far from facilitating fresh triumphs in our ownage, actually increase the difficulty of fresh triumphs by rendering originalityharder of attainment; not only is artistic achievement not cumulative, but itseems even to depend upon a certain freshness and naïveté of impulse andvision which civilisation tends to destroy. Hence comes, to those who havebeen nourished on the literary and artistic productions of former ages, a certainpeevishness and undue fastidiousness towards the present, from which thereseems no escape except into the deliberate vandalism which ignores traditionand in the search after originality achieves only the eccentric. But in suchvandalism there is none of the simplicity and spontaneity out of which great artsprings: theory is still the canker in its core, and insincerity destroys theadvantages of a merely pretended ignorance. The despair thus arising from an education which suggests no pre-eminentmental activity except that of artistic creation is wholly absent from aneducation [41]which gives the knowledge of scientific method. The discovery ofscientific method, except in pure mathematics, is a thing of yesterday; speakingbroadly, we may say that it dates from Galileo. Yet already it has transformedthe world, and its success proceeds with ever-accelerating velocity. In sciencemen have discovered an activity of the very highest value in which they are nolonger, as in art, dependent for progress upon the appearance of continuallygreater genius, for in science the successors stand upon the shoulders of theirpredecessors; where one man of supreme genius has invented a method, athousand lesser men can apply it. No transcendent ability is required in order tomake useful discoveries in science; the edifice of science needs its masons,bricklayers, and common labourers as well as its foremen, master-builders, andarchitects. In art nothing worth doing can be done without genius; in scienceeven a very moderate capacity can contribute to a supreme achievement. In science the man of real genius is the man who invents a new method. Thenotable discoveries are often made by his successors, who can apply themethod with fresh vigour, unimpaired by the previous labour of perfecting it;but the mental calibre of the thought required for their work, however brilliant,is not so great as that required by the first inventor of the method. There are inscience immense numbers of different methods, appropriate to different classesof problems; but over and above them all, there is something not easilydefinable, which may be called the method of science. It was formerlycustomary to identify this with the inductive method, and to associate it withthe name of Bacon. But the true inductive method was not discovered byBacon, and the true method of science [42]is something which includesdeduction as much as induction, logic and mathematics as much as botany andgeology. I shall not attempt the difficult task of stating what the scientificmethod is, but I will try to indicate the temper of mind out of which thescientific method grows, which is the second of the two merits that werementioned above as belonging to a scientific education. The kernel of the scientific outlook is a thing so simple, so obvious, soseemingly trivial, that the mention of it may almost excite derision. The kernelof the scientific outlook is the refusal to regard our own desires, tastes, andinterests as affording a key to the understanding of the world. Stated thusbaldly, this may seem no more than a trite truism. But to remember itconsistently in matters arousing our passionate partisanship is by no meanseasy, especially where the available evidence is uncertain and inconclusive. Afew illustrations will make this clear. Aristotle, I understand, considered that the stars must move in circles becausethe circle is the most perfect curve. In the absence of evidence to the contrary,he allowed himself to decide a question of fact by an appeal to æsthetico-moralconsiderations. In such a case it is at once obvious to us that this appeal wasunjustifiable. We know now how to ascertain as a fact the way in which theheavenly bodies move, and we know that they do not move in circles, or evenin accurate ellipses, or in any other kind of simply describable curve. This maybe painful to a certain hankering after simplicity of pattern in the universe, butwe know that in astronomy such feelings are irrelevant. Easy as this knowledgeseems now, we owe it to the courage and insight of the first inventors ofscientific method, and more especially of Galileo. [43]We may take as another illustration Malthus's doctrine of population. Thisillustration is all the better for the fact that his actual doctrine is now known tobe largely erroneous. It is not his conclusions that are valuable, but the temperand method of his inquiry. As everyone knows, it was to him that Darwin owedan essential part of his theory of natural selection, and this was only possiblebecause Malthus's outlook was truly scientific. His great merit lies inconsidering man not as the object of praise or blame, but as a part of nature, athing with a certain characteristic behaviour from which certain consequencesmust follow. If the behaviour is not quite what Malthus supposed, if theconsequences are not quite what he inferred, that may falsify his conclusions,but does not impair the value of his method. The objections which were madewhen his doctrine was new—that it was horrible and depressing, that peopleought not to act as he said they did, and so on—were all such as implied anunscientific attitude of mind; as against all of them, his calm determination totreat man as a natural phenomenon marks an important advance over thereformers of the eighteenth century and the Revolution. Under the influence of Darwinism the scientific attitude towards man hasnow become fairly common, and is to some people quite natural, though tomost it is still a difficult and artificial intellectual contortion. There is however,one study which is as yet almost wholly untouched by the scientific spirit—Imean the study of philosophy. Philosophers and the public imagine that thescientific spirit must pervade pages that bristle with allusions to ions, germ-plasms, and the eyes of shell-fish. But as the devil can quote Scripture, so thephilosopher can quote science. The scientific spirit is not an affairof [44]quotation, of externally acquired information, any more than manners arean affair of the etiquette-book. The scientific attitude of mind involves asweeping away of all other desires in the interests of the desire to know—itinvolves suppression of hopes and fears, loves and hates, and the wholesubjective emotional life, until we become subdued to the material, able to seeit frankly, without preconceptions, without bias, without any wish except to seeit as it is, and without any belief that what it is must be determined by somerelation, positive or negative, to what we should like it to be, or to what we caneasily imagine it to be. Now in philosophy this attitude of mind has not as yet been achieved. Acertain self-absorption, not personal, but human, has marked almost allattempts to conceive the universe as a whole. Mind, or some aspect of it—thought or will or sentience—has been regarded as the pattern after which theuniverse is to be conceived, for no better reason, at bottom, than that such auniverse would not seem strange, and would give us the cosy feeling that everyplace is like home. To conceive the universe as essentially progressive oressentially deteriorating, for example, is to give to our hopes and fears a cosmicimportance which may, of course, be justified, but which we have as yet noreason to suppose justified. Until we have learnt to think of it in ethicallyneutral terms, we have not arrived at a scientific attitude in philosophy; anduntil we have arrived at such an attitude, it is hardly to be hoped thatphilosophy will achieve any solid results. I have spoken so far largely of the negative aspect of the scientific spirit, butit is from the positive aspect that its value is derived. The instinct ofconstructiveness, which is one of the chief incentives to artistic creation, canfind [45]in scientific systems a satisfaction more massive than any epic poem.Disinterested curiosity, which is the source of almost all intellectual effort,finds with astonished delight that science can unveil secrets which might wellhave seemed for ever undiscoverable. The desire for a larger life and widerinterests, for an escape from private circumstances, and even from the wholerecurring human cycle of birth and death, is fulfilled by the impersonal cosmicoutlook of science as by nothing else. To all these must be added, ascontributing to the happiness of the man of science, the admiration of splendidachievement, and the consciousness of inestimable utility to the human race. Alife devoted to science is therefore a happy life, and its happiness is derivedfrom the very best sources that are open to dwellers on this troubled andpassionate planet.[46]

IIIToC

A FREE MAN'S WORSHIP[9]

To Dr. Faustus in his study Mephistopheles told the history of the Creation,saying: "The endless praises of the choirs of angels had begun to grow wearisome;for, after all, did he not deserve their praise? Had he not given them endlessjoy? Would it not be more amusing to obtain undeserved praise, to beworshipped by beings whom he tortured? He smiled inwardly, and resolvedthat the great drama should be performed. "For countless ages the hot nebula whirled aimlessly through space. Atlength it began to take shape, the central mass threw off planets, the planetscooled, boiling seas and burning mountains heaved and tossed, from blackmasses of cloud hot sheets of rain deluged the barely solid crust. And now thefirst germ of life grew in the depths of the ocean, and developed rapidly in thefructifying warmth into vast forest trees, huge ferns springing from the dampmould, sea monsters breeding, fighting, devouring, and passing away. Andfrom the monsters, as the play unfolded itself, Man was born, with the power ofthought, the knowledge of good and evil, and the cruel thirst for worship. AndMan saw that all is passing in this mad, monstrous world, that all is strugglingto snatch, at any cost, a few brief moments of life before Death's inexorabledecree. And [47]Man said: 'There is a hidden purpose, could we but fathom it,and the purpose is good; for we must reverence something, and in the visibleworld there is nothing worthy of reverence.' And Man stood aside from thestruggle, resolving that God intended harmony to come out of chaos by humanefforts. And when he followed the instincts which God had transmitted to himfrom his ancestry of beasts of prey, he called it Sin, and asked God to forgivehim. But he doubted whether he could be justly forgiven, until he invented adivine Plan by which God's wrath was to have been appeased. And seeing thepresent was bad, he made it yet worse, that thereby the future might be better.And he gave God thanks for the strength that enabled him to forgo even thejoys that were possible. And God smiled; and when he saw that Man hadbecome perfect in renunciation and worship, he sent another sun through thesky, which crashed into Man's sun; and all returned again to nebula. "'Yes,' he murmured, 'it was a good play; I will have it performed again.'" Such, in outline, but even more purposeless, more void of meaning, is theworld which Science presents for our belief. Amid such a world, if anywhere,our ideals henceforward must find a home. That Man is the product of causeswhich had no prevision of the end they were achieving; that his origin, hisgrowth, his hopes and fears, his loves and his beliefs, are but the outcome ofaccidental collocations of atoms; that no fire, no heroism, no intensity ofthought and feeling, can preserve an individual life beyond the grave; that allthe labours of the ages, all the devotion, all the inspiration, all the noondaybrightness of human genius, are destined to extinction in the vast death of thesolar [48]system, and that the whole temple of Man's achievement mustinevitably be buried beneath the débris of a universe in ruins—all these things,if not quite beyond dispute, are yet so nearly certain, that no philosophy whichrejects them can hope to stand. Only within the scaffolding of these truths, onlyon the firm foundation of unyielding despair, can the soul's habitationhenceforth be safely built. How, in such an alien and inhuman world, can so powerless a creature asMan preserve his aspirations untarnished? A strange mystery it is that Nature,omnipotent but blind, in the revolutions of her secular hurryings through theabysses of space, has brought forth at last a child, subject still to her power, butgifted with sight, with knowledge of good and evil, with the capacity of judgingall the works of his unthinking Mother. In spite of Death, the mark and seal ofthe parental control, Man is yet free, during his brief years, to examine, tocriticise, to know, and in imagination to create. To him alone, in the world withwhich he is acquainted, this freedom belongs; and in this lies his superiority tothe resistless forces that control his outward life. The savage, like ourselves, feels the oppression of his impotence before thepowers of Nature; but having in himself nothing that he respects more thanPower, he is willing to prostrate himself before his gods, without inquiringwhether they are worthy of his worship. Pathetic and very terrible is the longhistory of cruelty and torture, of degradation and human sacrifice, endured inthe hope of placating the jealous gods: surely, the trembling believer thinks,when what is most precious has been freely given, their lust for blood must beappeased, and more will not be required. The religion of [49]Moloch—as suchcreeds may be generically called—is in essence the cringing submission of theslave, who dare not, even in his heart, allow the thought that his masterdeserves no adulation. Since the independence of ideals is not yetacknowledged, Power may be freely worshipped, and receive an unlimitedrespect, despite its wanton infliction of pain. But gradually, as morality grows bolder, the claim of the ideal world beginsto be felt; and worship, if it is not to cease, must be given to gods of anotherkind than those created by the savage. Some, though they feel the demands ofthe ideal, will still consciously reject them, still urging that naked Power isworthy of worship. Such is the attitude inculcated in God's answer to Job out ofthe whirlwind: the divine power and knowledge are paraded, but of the divinegoodness there is no hint. Such also is the attitude of those who, in our ownday, base their morality upon the struggle for survival, maintaining that thesurvivors are necessarily the fittest. But others, not content with an answer sorepugnant to the moral sense, will adopt the position which we have becomeaccustomed to regard as specially religious, maintaining that, in some hiddenmanner, the world of fact is really harmonious with the world of ideals. ThusMan creates God, all-powerful and all-good, the mystic unity of what is andwhat should be. But the world of fact, after all, is not good; and, in submitting our judgmentto it, there is an element of slavishness from which our thoughts must bepurged. For in all things it is well to exalt the dignity of Man, by freeing him asfar as possible from the tyranny of non-human Power. When we have realisedthat Power is largely bad, that man, with his knowledge of good and evil, is buta helpless atom in a world which has no such [50]knowledge, the choice is againpresented to us: Shall we worship Force, or shall we worship Goodness? Shallour God exist and be evil, or shall he be recognised as the creation of our ownconscience? The answer to this question is very momentous, and affects profoundly ourwhole morality. The worship of Force, to which Carlyle and Nietzsche and thecreed of Militarism have accustomed us, is the result of failure to maintain ourown ideals against a hostile universe: it is itself a prostrate submission to evil, asacrifice of our best to Moloch. If strength indeed is to be respected, let usrespect rather the strength of those who refuse that false "recognition of facts"which fails to recognise that facts are often bad. Let us admit that, in the worldwe know, there are many things that would be better otherwise, and that theideals to which we do and must adhere are not realised in the realm of matter.Let us preserve our respect for truth, for beauty, for the ideal of perfectionwhich life does not permit us to attain, though none of these things meet withthe approval of the unconscious universe. If Power is bad, as it seems to be, letus reject it from our hearts. In this lies Man's true freedom: in determination toworship only the God created by our own love of the good, to respect only theheaven which inspires the insight of our best moments. In action, in desire, wemust submit perpetually to the tyranny of outside forces; but in thought, inaspiration, we are free, free from our fellow-men, free from the petty planet onwhich our bodies impotently crawl, free even, while we live, from the tyrannyof death. Let us learn, then, that energy of faith which enables us to liveconstantly in the vision of the good; and let us descend, in action, into theworld of fact, with that vision always before us. [51]When first the opposition of fact and ideal grows fully visible, a spirit offiery revolt, of fierce hatred of the gods, seems necessary to the assertion offreedom. To defy with Promethean constancy a hostile universe, to keep its evilalways in view, always actively hated, to refuse no pain that the malice ofPower can invent, appears to be the duty of all who will not bow before theinevitable. But indignation is still a bondage, for it compels our thoughts to beoccupied with an evil world; and in the fierceness of desire from whichrebellion springs there is a kind of self-assertion which it is necessary for thewise to overcome. Indignation is a submission of our thoughts, but not of ourdesires; the Stoic freedom in which wisdom consists is found in the submissionof our desires, but not of our thoughts. From the submission of our desiressprings the virtue of resignation; from the freedom of our thoughts springs thewhole world of art and philosophy, and the vision of beauty by which, at last,we half reconquer the reluctant world. But the vision of beauty is possible onlyto unfettered contemplation, to thoughts not weighted by the load of eagerwishes; and thus Freedom comes only to those who no longer ask of life that itshall yield them any of those personal goods that are subject to the mutations ofTime. Although the necessity of renunciation is evidence of the existence of evil,yet Christianity, in preaching it, has shown a wisdom exceeding that of thePromethean philosophy of rebellion. It must be admitted that, of the things wedesire, some, though they prove impossible, are yet real goods; others,however, as ardently longed for, do not form part of a fully purified ideal. Thebelief that what must be renounced is bad, though sometimes false, is far lessoften false than untamed passion supposes; and the creed of religion, byproviding a reason [52]for proving that it is never false, has been the means ofpurifying our hopes by the discovery of many austere truths. But there is in resignation a further good element: even real goods, when theyare unattainable, ought not to be fretfully desired. To every man comes, sooneror later, the great renunciation. For the young, there is nothing unattainable; agood thing desired with the whole force of a passionate will, and yetimpossible, is to them not credible. Yet, by death, by illness, by poverty, or bythe voice of duty, we must learn, each one of us, that the world was not madefor us, and that, however beautiful may be the things we crave, Fate maynevertheless forbid them. It is the part of courage, when misfortune comes, tobear without repining the ruin of our hopes, to turn away our thoughts fromvain regrets. This degree of submission to Power is not only just and right: it isthe very gate of wisdom. But passive renunciation is not the whole of wisdom; for not by renunciationalone can we build a temple for the worship of our own ideals. Hauntingforeshadowings of the temple appear in the realm of imagination, in music, inarchitecture, in the untroubled kingdom of reason, and in the golden sunsetmagic of lyrics, where beauty shines and glows, remote from the touch ofsorrow, remote from the fear of change, remote from the failures anddisenchantments of the world of fact. In the contemplation of these things thevision of heaven will shape itself in our hearts, giving at once a touchstone tojudge the world about us, and an inspiration by which to fashion to our needswhatever is not incapable of serving as a stone in the sacred temple. Except for those rare spirits that are born without sin, there is a cavern ofdarkness to be traversed before that [53]temple can be entered. The gate of thecavern is despair, and its floor is paved with the gravestones of abandonedhopes. There Self must die; there the eagerness, the greed of untamed desiremust be slain, for only so can the soul be freed from the empire of Fate. But outof the cavern the Gate of Renunciation leads again to the daylight of wisdom,by whose radiance a new insight, a new joy, a new tenderness, shine forth togladden the pilgrim's heart. When, without the bitterness of impotent rebellion, we have learnt both toresign ourselves to the outward rule of Fate and to recognise that the non-human world is unworthy of our worship, it becomes possible at last so totransform and refashion the unconscious universe, so to transmute it in thecrucible of imagination, that a new image of shining gold replaces the old idolof clay. In all the multiform facts of the world—in the visual shapes of treesand mountains and clouds, in the events of the life of man, even in the veryomnipotence of Death—the insight of creative idealism can find the reflectionof a beauty which its own thoughts first made. In this way mind asserts itssubtle mastery over the thoughtless forces of Nature. The more evil the materialwith which it deals, the more thwarting to untrained desire, the greater is itsachievement in inducing the reluctant rock to yield up its hidden treasures, theprouder its victory in compelling the opposing forces to swell the pageant of itstriumph. Of all the arts, Tragedy is the proudest, the most triumphant; for itbuilds its shining citadel in the very centre of the enemy's country, on the verysummit of his highest mountain; from its impregnable watchtowers, his campsand arsenals, his columns and forts, are all revealed; within its walls the freelife continues, while the legions of Death and Pain and Despair, and all [54]theservile captains of tyrant Fate, afford the burghers of that dauntless city newspectacles of beauty. Happy those sacred ramparts, thrice happy the dwellers onthat all-seeing eminence. Honour to those brave warriors who, throughcountless ages of warfare, have preserved for us the priceless heritage ofliberty, and have kept undefiled by sacrilegious invaders the home of theunsubdued. But the beauty of Tragedy does but make visible a quality which, in more orless obvious shapes, is present always and everywhere in life. In the spectacleof Death, in the endurance of intolerable pain, and in the irrevocableness of avanished past, there is a sacredness, an overpowering awe, a feeling of thevastness, the depth, the inexhaustible mystery of existence, in which, as bysome strange marriage of pain, the sufferer is bound to the world by bonds ofsorrow. In these moments of insight, we lose all eagerness of temporary desire,all struggling and striving for petty ends, all care for the little trivial things that,to a superficial view, make up the common life of day by day; we see,surrounding the narrow raft illumined by the flickering light of humancomradeship, the dark ocean on whose rolling waves we toss for a brief hour;from the great night without, a chill blast breaks in upon our refuge; all theloneliness of humanity amid hostile forces is concentrated upon the individualsoul, which must struggle alone, with what of courage it can command, againstthe whole weight of a universe that cares nothing for its hopes and fears.Victory, in this struggle with the powers of darkness, is the true baptism intothe glorious company of heroes, the true initiation into the overmasteringbeauty of human existence. From that awful encounter of the soul with theouter world, enunciation, wisdom, and charity are born; and with [55]their birtha new life begins. To take into the inmost shrine of the soul the irresistibleforces whose puppets we seem to be—Death and change, the irrevocableness ofthe past, and the powerlessness of man before the blind hurry of the universefrom vanity to vanity—to feel these things and know them is to conquer them. This is the reason why the Past has such magical power. The beauty of itsmotionless and silent pictures is like the enchanted purity of late autumn, whenthe leaves, though one breath would make them fall, still glow against the skyin golden glory. The Past does not change or strive; like Duncan, after life'sfitful fever it sleeps well; what was eager and grasping, what was petty andtransitory, has faded away, the things that were beautiful and eternal shine outof it like stars in the night. Its beauty, to a soul not worthy of it, is unendurable;but to a soul which has conquered Fate it is the key of religion. The life of Man, viewed outwardly, is but a small thing in comparison withthe forces of Nature. The slave is doomed to worship Time and Fate and Death,because they are greater than anything he finds in himself, and because all histhoughts are of things which they devour. But, great as they are, to think ofthem greatly, to feel their passionless splendour, is greater still. And suchthought makes us free men; we no longer bow before the inevitable in Orientalsubjection, but we absorb it, and make it a part of ourselves. To abandon thestruggle for private happiness, to expel all eagerness of temporary desire, toburn with passion for eternal things—this is emancipation, and this is the freeman's worship. And this liberation is effected by a contemplation of Fate; forFate itself is subdued by the [56]mind which leaves nothing to be purged by thepurifying fire of Time. United with his fellow-men by the strongest of all ties, the tie of a commondoom, the free man finds that a new vision is with him always, shedding overevery daily task the light of love. The life of Man is a long march through thenight, surrounded by invisible foes, tortured by weariness and pain, towards agoal that few can hope to reach, and where none may tarry long. One by one, asthey march, our comrades vanish from our sight, seized by the silent orders ofomnipotent Death. Very brief is the time in which we can help them, in whichtheir happiness or misery is decided. Be it ours to shed sunshine on their path,to lighten their sorrows by the balm of sympathy, to give them the pure joy of anever-tiring affection, to strengthen failing courage, to instil faith in hours ofdespair. Let us not weigh in grudging scales their merits and demerits, but let usthink only of their need—of the sorrows, the difficulties, perhaps theblindnesses, that make the misery of their lives; let us remember that they arefellow-sufferers in the same darkness, actors in the same tragedy withourselves. And so, when their day is over, when their good and their evil havebecome eternal by the immortality of the past, be it ours to feel that, where theysuffered, where they failed, no deed of ours was the cause; but wherever aspark of the divine fire kindled in their hearts, we were ready withencouragement, with sympathy, with brave words in which high courageglowed. Brief and powerless is Man's life; on him and all his race the slow, sure doomfalls pitiless and dark. Blind to good and evil, reckless of destruction,omnipotent matter rolls on its relentless way; for Man, condemned to-day tolose his dearest, to-morrow himself to pass [57]through the gate of darkness, itremains only to cherish, ere yet the blow falls, the lofty thoughts that ennoblehis little day; disdaining the coward terrors of the slave of Fate, to worship atthe shrine that his own hands have built; undismayed by the empire of chance,to preserve a mind free from the wanton tyranny that rules his outward life;proudly defiant of the irresistible forces that tolerate, for a moment, hisknowledge and his condemnation, to sustain alone, a weary but unyieldingAtlas, the world that his own ideals have fashioned despite the trampling marchof unconscious power.

FOOTNOTES:

[9]Reprinted from the Independent Review, December, 1903.

[58]

IVToC

THE STUDY OF MATHEMATICS

In regard to every form of human activity it is necessary that the question

should be asked from time to time, What is its purpose and ideal? In what waydoes it contribute to the beauty of human existence? As respects those pursuitswhich contribute only remotely, by providing the mechanism of life, it is wellto be reminded that not the mere fact of living is to be desired, but the art ofliving in the contemplation of great things. Still more in regard to thoseavocations which have no end outside themselves, which are to be justified, ifat all, as actually adding to the sum of the world's permanent possessions, it isnecessary to keep alive a knowledge of their aims, a clear prefiguring vision ofthe temple in which creative imagination is to be embodied. The fulfilment of this need, in what concerns the studies forming the materialupon which custom has decided to train the youthful mind, is indeed sadlyremote—so remote as to make the mere statement of such a claim appearpreposterous. Great men, fully alive to the beauty of the contemplations towhose service their lives are devoted, desiring that others may share in theirjoys, persuade mankind to impart to the successive generations the mechanicalknowledge without which it is impossible to cross the threshold. Dry pedantspossess themselves of the privilege of instilling this knowledge: they forget thatit is to serve but as a [59]key to open the doors of the temple; though they spendtheir lives on the steps leading up to those sacred doors, they turn their backsupon the temple so resolutely that its very existence is forgotten, and the eageryouth, who would press forward to be initiated to its domes and arches, isbidden to turn back and count the steps. Mathematics, perhaps more even than the study of Greece and Rome, hassuffered from this oblivion of its due place in civilisation. Although traditionhas decreed that the great bulk of educated men shall know at least the elementsof the subject, the reasons for which the tradition arose are forgotten, buriedbeneath a great rubbish-heap of pedantries and trivialities. To those who inquireas to the purpose of mathematics, the usual answer will be that it facilitates themaking of machines, the travelling from place to place, and the victory overforeign nations, whether in war or commerce. If it be objected that these ends—all of which are of doubtful value—are not furthered by the merely elementarystudy imposed upon those who do not become expert mathematicians, thereply, it is true, will probably be that mathematics trains the reasoning faculties.Yet the very men who make this reply are, for the most part, unwilling toabandon the teaching of definite fallacies, known to be such, and instinctivelyrejected by the unsophisticated mind of every intelligent learner. And thereasoning faculty itself is generally conceived, by those who urge itscultivation, as merely a means for the avoidance of pitfalls and a help in thediscovery of rules for the guidance of practical life. All these are undeniablyimportant achievements to the credit of mathematics; yet it is none of these thatentitles mathematics to a place in every liberal education. Plato, we know,regarded the contemplation of mathematical truths as worthy of the [60]Deity;and Plato realised, more perhaps than any other single man, what thoseelements are in human life which merit a place in heaven. There is inmathematics, he says, "something which is necessary and cannot be set aside ...and, if I mistake not, of divine necessity; for as to the human necessities ofwhich the Many talk in this connection, nothing can be more ridiculous thansuch an application of the words. Cleinias. And what are these necessities ofknowledge, Stranger, which are divine and not human? Athenian. Those thingswithout some use or knowledge of which a man cannot become a God to theworld, nor a spirit, nor yet a hero, nor able earnestly to think and care for man"(Laws, p. 818).[10] Such was Plato's judgment of mathematics; but themathematicians do not read Plato, while those who read him know nomathematics, and regard his opinion upon this question as merely a curiousaberration. Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part ofour weaker nature, without the gorgeous trappings of painting or music, yetsublimely pure, and capable of a stern perfection such as only the greatest artcan show. The true spirit of delight, the exaltation, the sense of being more thanman, which is the touchstone of the highest excellence, is to be found inmathematics as surely as in poetry. What is best in mathematics deserves notmerely to be learnt as a task, but to be assimilated as a part of daily thought,and brought again and again before the mind with ever-renewedencouragement. Real life is, to most men, a long second-best, a perpetualcompromise between the ideal and the possible; but the world of pure reasonknows no compromise, no practical [61]limitations, no barrier to the creativeactivity embodying in splendid edifices the passionate aspiration after theperfect from which all great work springs. Remote from human passions,remote even from the pitiful facts of nature, the generations have graduallycreated an ordered cosmos, where pure thought can dwell as in its naturalhome, and where one, at least, of our nobler impulses can escape from thedreary exile of the actual world. So little, however, have mathematicians aimed at beauty, that hardlyanything in their work has had this conscious purpose. Much, owing toirrepressible instincts, which were better than avowed beliefs, has beenmoulded by an unconscious taste; but much also has been spoilt by falsenotions of what was fitting. The characteristic excellence of mathematics isonly to be found where the reasoning is rigidly logical: the rules of logic are tomathematics what those of structure are to architecture. In the most beautifulwork, a chain of argument is presented in which every link is important on itsown account, in which there is an air of ease and lucidity throughout, and thepremises achieve more than would have been thought possible, by means whichappear natural and inevitable. Literature embodies what is general in particularcircumstances whose universal significance shines through their individualdress; but mathematics endeavours to present whatever is most general in itspurity, without any irrelevant trappings. How should the teaching of mathematics be conducted so as to communicateto the learner as much as possible of this high ideal? Here experience must, in agreat measure, be our guide; but some maxims may result from ourconsideration of the ultimate purpose to be achieved. [62]Oneof the chief ends served by mathematics, when rightly taught, is toawaken the learner's belief in reason, his confidence in the truth of what hasbeen demonstrated, and in the value of demonstration. This purpose is notserved by existing instruction; but it is easy to see ways in which it might beserved. At present, in what concerns arithmetic, the boy or girl is given a set ofrules, which present themselves as neither true nor false, but as merely the willof the teacher, the way in which, for some unfathomable reason, the teacherprefers to have the game played. To some degree, in a study of such definitepractical utility, this is no doubt unavoidable; but as soon as possible, thereasons of rules should be set forth by whatever means most readily appeal tothe childish mind. In geometry, instead of the tedious apparatus of fallaciousproofs for obvious truisms which constitutes the beginning of Euclid, thelearner should be allowed at first to assume the truth of everything obvious, andshould be instructed in the demonstrations of theorems which are at oncestartling and easily verifiable by actual drawing, such as those in which it isshown that three or more lines meet in a point. In this way belief is generated; itis seen that reasoning may lead to startling conclusions, which nevertheless thefacts will verify; and thus the instinctive distrust of whatever is abstract orrational is gradually overcome. Where theorems are difficult, they should befirst taught as exercises in geometrical drawing, until the figure has becomethoroughly familiar; it will then be an agreeable advance to be taught thelogical connections of the various lines or circles that occur. It is desirable alsothat the figure illustrating a theorem should be drawn in all possible cases andshapes, that so the abstract relations with which geometry is concerned may ofthemselves [63]emerge as the residue of similarity amid such great apparentdiversity. In this way the abstract demonstrations should form but a small partof the instruction, and should be given when, by familiarity with concreteillustrations, they have come to be felt as the natural embodiment of visiblefact. In this early stage proofs should not be given with pedantic fullness;definitely fallacious methods, such as that of superposition, should be rigidlyexcluded from the first, but where, without such methods, the proof would bevery difficult, the result should be rendered acceptable by arguments andillustrations which are explicitly contrasted with demonstrations. In the beginning of algebra, even the most intelligent child finds, as a rule,very great difficulty. The use of letters is a mystery, which seems to have nopurpose except mystification. It is almost impossible, at first, not to think thatevery letter stands for some particular number, if only the teacher wouldreveal what number it stands for. The fact is, that in algebra the mind is firsttaught to consider general truths, truths which are not asserted to hold only ofthis or that particular thing, but of any one of a whole group of things. It is inthe power of understanding and discovering such truths that the mastery of theintellect over the whole world of things actual and possible resides; and abilityto deal with the general as such is one of the gifts that a mathematical educationshould bestow. But how little, as a rule, is the teacher of algebra able to explainthe chasm which divides it from arithmetic, and how little is the learner assistedin his groping efforts at comprehension! Usually the method that has beenadopted in arithmetic is continued: rules are set forth, with no adequateexplanation of their grounds; the pupil learns to use the rules blindly, [64]andpresently, when he is able to obtain the answer that the teacher desires, he feelsthat he has mastered the difficulties of the subject. But of inner comprehensionof the processes employed he has probably acquired almost nothing. When algebra has been learnt, all goes smoothly until we reach those studiesin which the notion of infinity is employed—the infinitesimal calculus and thewhole of higher mathematics. The solution of the difficulties which formerlysurrounded the mathematical infinite is probably the greatest achievement ofwhich our own age has to boast. Since the beginnings of Greek thought thesedifficulties have been known; in every age the finest intellects have vainlyendeavoured to answer the apparently unanswerable questions that had beenasked by Zeno the Eleatic. At last Georg Cantor has found the answer, and hasconquered for the intellect a new and vast province which had been given overto Chaos and old Night. It was assumed as self-evident, until Cantor andDedekind established the opposite, that if, from any collection of things, somewere taken away, the number of things left must always be less than theoriginal number of things. This assumption, as a matter of fact, holds only offinite collections; and the rejection of it, where the infinite is concerned, hasbeen shown to remove all the difficulties that had hitherto baffled humanreason in this matter, and to render possible the creation of an exact science ofthe infinite. This stupendous fact ought to produce a revolution in the higherteaching of mathematics; it has itself added immeasurably to the educationalvalue of the subject, and it has at last given the means of treating with logicalprecision many studies which, until lately, were wrapped in fallacy andobscurity. By those who were educated on the [65]old lines, the new work isconsidered to be appallingly difficult, abstruse, and obscure; and it must beconfessed that the discoverer, as is so often the case, has hardly himselfemerged from the mists which the light of his intellect is dispelling. Butinherently, the new doctrine of the infinite, to all candid and inquiring minds,has facilitated the mastery of higher mathematics; for hitherto, it has beennecessary to learn, by a long process of sophistication, to give assent toarguments which, on first acquaintance, were rightly judged to be confused anderroneous. So far from producing a fearless belief in reason, a bold rejection ofwhatever failed to fulfil the strictest requirements of logic, a mathematicaltraining, during the past two centuries, encouraged the belief that many things,which a rigid inquiry would reject as fallacious, must yet be accepted becausethey work in what the mathematician calls "practice." By this means, a timid,compromising spirit, or else a sacerdotal belief in mysteries not intelligible tothe profane, has been bred where reason alone should have ruled. All this it isnow time to sweep away; let those who wish to penetrate into the arcana ofmathematics be taught at once the true theory in all its logical purity, and in theconcatenation established by the very essence of the entities concerned. If we are considering mathematics as an end in itself, and not as a technicaltraining for engineers, it is very desirable to preserve the purity and strictness ofits reasoning. Accordingly those who have attained a sufficient familiarity withits easier portions should be led backward from propositions to which they haveassented as self-evident to more and more fundamental principles from whichwhat had previously appeared as premises can be deduced. They shouldbe [66]taught—what the theory of infinity very aptly illustrates—that manypropositions seem self-evident to the untrained mind which, nevertheless, anearer scrutiny shows to be false. By this means they will be led to a scepticalinquiry into first principles, an examination of the foundations upon which thewhole edifice of reasoning is built, or, to take perhaps a more fitting metaphor,the great trunk from which the spreading branches spring. At this stage, it iswell to study afresh the elementary portions of mathematics, asking no longermerely whether a given proposition is true, but also how it grows out of thecentral principles of logic. Questions of this nature can now be answered with aprecision and certainty which were formerly quite impossible; and in the chainsof reasoning that the answer requires the unity of all mathematical studies atlast unfolds itself. In the great majority of mathematical text-books there is a total lack of unityin method and of systematic development of a central theme. Propositions ofvery diverse kinds are proved by whatever means are thought most easilyintelligible, and much space is devoted to mere curiosities which in no waycontribute to the main argument. But in the greatest works, unity andinevitability are felt as in the unfolding of a drama; in the premisses a subject isproposed for consideration, and in every subsequent step some definite advanceis made towards mastery of its nature. The love of system, of interconnection,which is perhaps the inmost essence of the intellectual impulse, can find freeplay in mathematics as nowhere else. The learner who feels this impulse mustnot be repelled by an array of meaningless examples or distracted by amusingoddities, but must be encouraged to dwell upon central principles, to becomefamiliar with the structure of the various subjects which are put before [67]him,to travel easily over the steps of the more important deductions. In this way agood tone of mind is cultivated, and selective attention is taught to dwell bypreference upon what is weighty and essential. When the separate studies into which mathematics is divided have each beenviewed as a logical whole, as a natural growth from the propositions whichconstitute their principles, the learner will be able to understand thefundamental science which unifies and systematises the whole of deductivereasoning. This is symbolic logic—a study which, though it owes its inceptionto Aristotle, is yet, in its wider developments, a product, almost wholly, of thenineteenth century, and is indeed, in the present day, still growing with greatrapidity. The true method of discovery in symbolic logic, and probably also thebest method for introducing the study to a learner acquainted with other parts ofmathematics, is the analysis of actual examples of deductive reasoning, with aview to the discovery of the principles employed. These principles, for the mostpart, are so embedded in our ratiocinative instincts, that they are employedquite unconsciously, and can be dragged to light only by much patient effort.But when at last they have been found, they are seen to be few in number, andto be the sole source of everything in pure mathematics. The discovery that allmathematics follows inevitably from a small collection of fundamental laws isone which immeasurably enhances the intellectual beauty of the whole; to thosewho have been oppressed by the fragmentary and incomplete nature of mostexisting chains of deduction this discovery comes with all the overwhelmingforce of a revelation; like a palace emerging from the autumn mist as thetraveller ascends an Italian hill-side, the stately storeys of the mathematicaledifice appear in their [68]due order and proportion, with a new perfection inevery part. Until symbolic logic had acquired its present development, the principlesupon which mathematics depends were always supposed to be philosophical,and discoverable only by the uncertain, unprogressive methods hithertoemployed by philosophers. So long as this was thought, mathematics seemed tobe not autonomous, but dependent upon a study which had quite other methodsthan its own. Moreover, since the nature of the postulates from whicharithmetic, analysis, and geometry are to be deduced was wrapped in all thetraditional obscurities of metaphysical discussion, the edifice built upon suchdubious foundations began to be viewed as no better than a castle in the air. Inthis respect, the discovery that the true principles are as much a part ofmathematics as any of their consequences has very greatly increased theintellectual satisfaction to be obtained. This satisfaction ought not to be refusedto learners capable of enjoying it, for it is of a kind to increase our respect forhuman powers and our knowledge of the beauties belonging to the abstractworld. Philosophers have commonly held that the laws of logic, which underliemathematics, are laws of thought, laws regulating the operations of our minds.By this opinion the true dignity of reason is very greatly lowered: it ceases tobe an investigation into the very heart and immutable essence of all thingsactual and possible, becoming, instead, an inquiry into something more or lesshuman and subject to our limitations. The contemplation of what is non-human,the discovery that our minds are capable of dealing with material not created bythem, above all, the realisation that beauty belongs to the outer world as to theinner, are the chief means of overcoming [69]the terrible sense of impotence, ofweakness, of exile amid hostile powers, which is too apt to result fromacknowledging the all-but omnipotence of alien forces. To reconcile us, by theexhibition of its awful beauty, to the reign of Fate—which is merely the literarypersonification of these forces—is the task of tragedy. But mathematics takesus still further from what is human, into the region of absolute necessity, towhich not only the actual world, but every possible world, must conform; andeven here it builds a habitation, or rather finds a habitation eternally standing,where our ideals are fully satisfied and our best hopes are not thwarted. It isonly when we thoroughly understand the entire independence of ourselves,which belongs to this world that reason finds, that we can adequately realise theprofound importance of its beauty. Not only is mathematics independent of us and our thoughts, but in anothersense we and the whole universe of existing things are independent ofmathematics. The apprehension of this purely ideal character is indispensable,if we are to understand rightly the place of mathematics as one among the arts.It was formerly supposed that pure reason could decide, in some respects, as tothe nature of the actual world: geometry, at least, was thought to deal with thespace in which we live. But we now know that pure mathematics can neverpronounce upon questions of actual existence: the world of reason, in a sense,controls the world of fact, but it is not at any point creative of fact, and in theapplication of its results to the world in time and space, its certainty andprecision are lost among approximations and working hypotheses. The objectsconsidered by mathematicians have, in the past, been mainly of a kindsuggested by phenomena; but from such restrictions the abstractimagination [70]should be wholly free. A reciprocal liberty must thus beaccorded: reason cannot dictate to the world of facts, but the facts cannotrestrict reason's privilege of dealing with whatever objects its love of beautymay cause to seem worthy of consideration. Here, as elsewhere, we build upour own ideals out of the fragments to be found in the world; and in the end it ishard to say whether the result is a creation or a discovery. It is very desirable, in instruction, not merely to persuade the student of theaccuracy of important theorems, but to persuade him in the way which itselfhas, of all possible ways, the most beauty. The true interest of a demonstrationis not, as traditional modes of exposition suggest, concentrated wholly in theresult; where this does occur, it must be viewed as a defect, to be remedied, ifpossible, by so generalising the steps of the proof that each becomes importantin and for itself. An argument which serves only to prove a conclusion is like astory subordinated to some moral which it is meant to teach: for æstheticperfection no part of the whole should be merely a means. A certain practicalspirit, a desire for rapid progress, for conquest of new realms, is responsible forthe undue emphasis upon results which prevails in mathematical instruction.The better way is to propose some theme for consideration—in geometry, afigure having important properties; in analysis, a function of which the study isilluminating, and so on. Whenever proofs depend upon some only of the marksby which we define the object to be studied, these marks should be isolated andinvestigated on their own account. For it is a defect, in an argument, to employmore premisses than the conclusion demands: what mathematicians callelegance results from employing only the essential principles in virtue of whichthe thesis is true. It is a merit in [71]Euclid that he advances as far as he is able togo without employing the axiom of parallels—not, as is often said, because thisaxiom is inherently objectionable, but because, in mathematics, every newaxiom diminishes the generality of the resulting theorems, and the greatestpossible generality is before all things to be sought. Of the effects of mathematics outside its own sphere more has been writtenthan on the subject of its own proper ideal. The effect upon philosophy has, inthe past, been most notable, but most varied; in the seventeenth century,idealism and rationalism, in the eighteenth, materialism and sensationalism,seemed equally its offspring. Of the effect which it is likely to have in thefuture it would be very rash to say much; but in one respect a good resultappears probable. Against that kind of scepticism which abandons the pursuitof ideals because the road is arduous and the goal not certainly attainable,mathematics, within its own sphere, is a complete answer. Too often it is saidthat there is no absolute truth, but only opinion and private judgment; that eachof us is conditioned, in his view of the world, by his own peculiarities, his owntaste and bias; that there is no external kingdom of truth to which, by patienceand discipline, we may at last obtain admittance, but only truth for me, for you,for every separate person. By this habit of mind one of the chief ends of humaneffort is denied, and the supreme virtue of candour, of fearless acknowledgmentof what is, disappears from our moral vision. Of such scepticism mathematicsis a perpetual reproof; for its edifice of truths stands unshakable andinexpungable to all the weapons of doubting cynicism. The effects of mathematics upon practical life, though they should not beregarded as the motive of our studies, may be used to answer a doubt to whichthe solitary[72]student must always be liable. In a world so full of evil andsuffering, retirement into the cloister of contemplation, to the enjoyment ofdelights which, however noble, must always be for the few only, cannot butappear as a somewhat selfish refusal to share the burden imposed upon othersby accidents in which justice plays no part. Have any of us the right, we ask, towithdraw from present evils, to leave our fellow-men unaided, while we live alife which, though arduous and austere, is yet plainly good in its own nature?When these questions arise, the true answer is, no doubt, that some must keepalive the sacred fire, some must preserve, in every generation, the hauntingvision which shadows forth the goal of so much striving. But when, as mustsometimes occur, this answer seems too cold, when we are almost maddenedby the spectacle of sorrows to which we bring no help, then we may reflect thatindirectly the mathematician often does more for human happiness than any ofhis more practically active contemporaries. The history of science abundantlyproves that a body of abstract propositions—even if, as in the case of conicsections, it remains two thousand years without effect upon daily life—mayyet, at any moment, be used to cause a revolution in the habitual thoughts andoccupations of every citizen. The use of steam and electricity—to take strikinginstances—is rendered possible only by mathematics. In the results of abstractthought the world possesses a capital of which the employment in enriching thecommon round has no hitherto discoverable limits. Nor does experience giveany means of deciding what parts of mathematics will be found useful. Utility,therefore, can be only a consolation in moments of discouragement, not a guidein directing our studies. For the health of the moral life, for ennobling the tone [73]of an age or anation, the austerer virtues have a strange power, exceeding the power of thosenot informed and purified by thought. Of these austerer virtues the love of truthis the chief, and in mathematics, more than elsewhere, the love of truth mayfind encouragement for waning faith. Every great study is not only an end initself, but also a means of creating and sustaining a lofty habit of mind; and thispurpose should be kept always in view throughout the teaching and learning ofmathematics.

FOOTNOTES:

[10]This passage was pointed out to me by Professor Gilbert Murray.

[74]

VToC

MATHEMATICS AND THE METAPHYSICIANS

The nineteenth century, which prided itself upon the invention of steam andevolution, might have derived a more legitimate title to fame from thediscovery of pure mathematics. This science, like most others, was baptisedlong before it was born; and thus we find writers before the nineteenth centuryalluding to what they called pure mathematics. But if they had been asked whatthis subject was, they would only have been able to say that it consisted ofArithmetic, Algebra, Geometry, and so on. As to what these studies had incommon, and as to what distinguished them from applied mathematics, ourancestors were completely in the dark. Pure mathematics was discovered by Boole, in a work which he calledthe Laws of Thought (1854). This work abounds in asseverations that it is notmathematical, the fact being that Boole was too modest to suppose his book thefirst ever written on mathematics. He was also mistaken in supposing that hewas dealing with the laws of thought: the question how people actually thinkwas quite irrelevant to him, and if his book had really contained the laws ofthought, it was curious that no one should ever have thought in such a waybefore. His book was in fact concerned with formal logic, and this is the samething as mathematics. [75]Puremathematics consists entirely of assertions to the effect that, if suchand such a proposition is true of anything, then such and such anotherproposition is true of that thing. It is essential not to discuss whether the firstproposition is really true, and not to mention what the anything is, of which it issupposed to be true. Both these points would belong to applied mathematics.We start, in pure mathematics, from certain rules of inference, by which we caninfer that if one proposition is true, then so is some other proposition. Theserules of inference constitute the major part of the principles of formal logic. Wethen take any hypothesis that seems amusing, and deduce itsconsequences. If our hypothesis is about anything, and not about some one ormore particular things, then our deductions constitute mathematics. Thusmathematics may be defined as the subject in which we never know what weare talking about, nor whether what we are saying is true. People who havebeen puzzled by the beginnings of mathematics will, I hope, find comfort inthis definition, and will probably agree that it is accurate. As one of the chief triumphs of modern mathematics consists in havingdiscovered what mathematics really is, a few more words on this subject maynot be amiss. It is common to start any branch of mathematics—for instance,Geometry—with a certain number of primitive ideas, supposed incapable ofdefinition, and a certain number of primitive propositions or axioms, supposedincapable of proof. Now the fact is that, though there are indefinables andindemonstrables in every branch of applied mathematics, there are none in puremathematics except such as belong to general logic. Logic, broadly speaking, isdistinguished by the fact that its propositions can be put into a form in whichthey apply to anything whatever. All pure mathematics—Arithmetic,Analysis, [76]and Geometry—is built up by combinations of the primitive ideasof logic, and its propositions are deduced from the general axioms of logic,such as the syllogism and the other rules of inference. And this is no longer adream or an aspiration. On the contrary, over the greater and more difficult partof the domain of mathematics, it has been already accomplished; in the fewremaining cases, there is no special difficulty, and it is now being rapidlyachieved. Philosophers have disputed for ages whether such deduction waspossible; mathematicians have sat down and made the deduction. For thephilosophers there is now nothing left but graceful acknowledgments. The subject of formal logic, which has thus at last shown itself to be identicalwith mathematics, was, as every one knows, invented by Aristotle, and formedthe chief study (other than theology) of the Middle Ages. But Aristotle nevergot beyond the syllogism, which is a very small part of the subject, and theschoolmen never got beyond Aristotle. If any proof were required of oursuperiority to the mediæval doctors, it might be found in this. Throughout theMiddle Ages, almost all the best intellects devoted themselves to formal logic,whereas in the nineteenth century only an infinitesimal proportion of theworld's thought went into this subject. Nevertheless, in each decade since 1850more has been done to advance the subject than in the whole period fromAristotle to Leibniz. People have discovered how to make reasoning symbolic,as it is in Algebra, so that deductions are effected by mathematical rules. Theyhave discovered many rules besides the syllogism, and a new branch of logic,called the Logic of Relatives,[11] has been invented to deal with topics thatwholly surpassed the powers of [77]the old logic, though they form the chiefcontents of mathematics. It is not easy for the lay mind to realise the importance of symbolism indiscussing the foundations of mathematics, and the explanation may perhapsseem strangely paradoxical. The fact is that symbolism is useful because itmakes things difficult. (This is not true of the advanced parts of mathematics,but only of the beginnings.) What we wish to know is, what can be deducedfrom what. Now, in the beginnings, everything is self-evident; and it is veryhard to see whether one self-evident proposition follows from another or not.Obviousness is always the enemy to correctness. Hence we invent some newand difficult symbolism, in which nothing seems obvious. Then we set upcertain rules for operating on the symbols, and the whole thing becomesmechanical. In this way we find out what must be taken as premiss and whatcan be demonstrated or defined. For instance, the whole of Arithmetic andAlgebra has been shown to require three indefinable notions and fiveindemonstrable propositions. But without a symbolism it would have been veryhard to find this out. It is so obvious that two and two are four, that we canhardly make ourselves sufficiently sceptical to doubt whether it can be proved.And the same holds in other cases where self-evident things are to be proved. But the proof of self-evident propositions may seem, to the uninitiated, asomewhat frivolous occupation. To this we might reply that it is often by nomeans self-evident that one obvious proposition follows from another obviousproposition; so that we are really discovering new truths when we prove what isevident by a method which is not evident. But a more interesting retort is, thatsince people have tried to prove obvious propositions, they have found thatmany of them are false. [78]Self-evidence is often a mere will-o'-the-wisp, whichis sure to lead us astray if we take it as our guide. For instance, nothing isplainer than that a whole always has more terms than a part, or that a number isincreased by adding one to it. But these propositions are now known to beusually false. Most numbers are infinite, and if a number is infinite you mayadd ones to it as long as you like without disturbing it in the least. One of themerits of a proof is that it instils a certain doubt as to the result proved; andwhen what is obvious can be proved in some cases, but not in others, itbecomes possible to suppose that in these other cases it is false. The great master of the art of formal reasoning, among the men of our ownday, is an Italian, Professor Peano, of the University of Turin.[12] He hasreduced the greater part of mathematics (and he or his followers will, in time,have reduced the whole) to strict symbolic form, in which there are no words atall. In the ordinary mathematical books, there are no doubt fewer words thanmost readers would wish. Still, little phrases occur, such as therefore, let usassume, consider, or hence it follows. All these, however, are a concession, andare swept away by Professor Peano. For instance, if we wish to learn the wholeof Arithmetic, Algebra, the Calculus, and indeed all that is usually called puremathematics (except Geometry), we must start with a dictionary of three words.One symbol stands for zero, another fornumber, and a third for next after. Whatthese ideas mean, it is necessary to know if you wish to become anarithmetician. But after symbols have been invented for these three ideas, notanother word is required in the whole development. All future symbols aresymbolically explained by means of these [79]three. Even these three can beexplained by means of the notions of relation and class; but this requires theLogic of Relations, which Professor Peano has never taken up. It must beadmitted that what a mathematician has to know to begin with is not much.There are at most a dozen notions out of which all the notions in all puremathematics (including Geometry) are compounded. Professor Peano, who isassisted by a very able school of young Italian disciples, has shown how thismay be done; and although the method which he has invented is capable ofbeing carried a good deal further than he has carried it, the honour of thepioneer must belong to him. Two hundred years ago, Leibniz foresaw the science which Peano hasperfected, and endeavoured to create it. He was prevented from succeeding byrespect for the authority of Aristotle, whom he could not believe guilty ofdefinite, formal fallacies; but the subject which he desired to create now exists,in spite of the patronising contempt with which his schemes have been treatedby all superior persons. From this "Universal Characteristic," as he called it, hehoped for a solution of all problems, and an end to all disputes. "Ifcontroversies were to arise," he says, "there would be no more need ofdisputation between two philosophers than between two accountants. For itwould suffice to take their pens in their hands, to sit down to their desks, and tosay to each other (with a friend as witness, if they liked), 'Let us calculate.'"This optimism has now appeared to be somewhat excessive; there still areproblems whose solution is doubtful, and disputes which calculation cannotdecide. But over an enormous field of what was formerly controversial,Leibniz's dream has become sober fact. In the whole philosophy ofmathematics, which [80]used to be at least as full of doubt as any other part ofphilosophy, order and certainty have replaced the confusion and hesitationwhich formerly reigned. Philosophers, of course, have not yet discovered thisfact, and continue to write on such subjects in the old way. But mathematicians,at least in Italy, have now the power of treating the principles of mathematics inan exact and masterly manner, by means of which the certainty of mathematicsextends also to mathematical philosophy. Hence many of the topics which usedto be placed among the great mysteries—for example, the natures of infinity, ofcontinuity, of space, time and motion—are now no longer in any degree opento doubt or discussion. Those who wish to know the nature of these things needonly read the works of such men as Peano or Georg Cantor; they will there findexact and indubitable expositions of all these quondam mysteries. In this capricious world, nothing is more capricious than posthumous fame.One of the most notable examples of posterity's lack of judgment is the EleaticZeno. This man, who may be regarded as the founder of the philosophy ofinfinity, appears in Plato's Parmenides in the privileged position of instructor toSocrates. He invented four arguments, all immeasurably subtle and profound,to prove that motion is impossible, that Achilles can never overtake the tortoise,and that an arrow in flight is really at rest. After being refuted by Aristotle, andby every subsequent philosopher from that day to our own, these argumentswere reinstated, and made the basis of a mathematical renaissance, by aGerman professor, who probably never dreamed of any connection betweenhimself and Zeno. Weierstrass,[13] by strictly [81]banishing from mathematicsthe use of infinitesimals, has at last shown that we live in an unchanging world,and that the arrow in its flight is truly at rest. Zeno's only error lay in inferring(if he did infer) that, because there is no such thing as a state of change,therefore the world is in the same state at any one time as at any other. This is aconsequence which by no means follows; and in this respect, the Germanmathematician is more constructive than the ingenious Greek. Weierstrass hasbeen able, by embodying his views in mathematics, where familiarity with trutheliminates the vulgar prejudices of common sense, to invest Zeno's paradoxeswith the respectable air of platitudes; and if the result is less delightful to thelover of reason than Zeno's bold defiance, it is at any rate more calculated toappease the mass of academic mankind. Zeno was concerned, as a matter of fact, with three problems, each presentedby motion, but each more abstract than motion, and capable of a purelyarithmetical treatment. These are the problems of the infinitesimal, the infinite,and continuity. To state clearly the difficulties involved, was to accomplishperhaps the hardest part of the philosopher's task. This was done by Zeno. Fromhim to our own day, the finest intellects of each generation in turn attacked theproblems, but achieved, broadly speaking, nothing. In our own time, however,three men—Weierstrass, Dedekind, and Cantor—have not merely advanced thethree problems, but have completely solved them. The solutions, for thoseacquainted with mathematics, are so clear as to leave no longer the slightestdoubt or difficulty. This achievement is probably the greatest of which our agehas to boast; and I know of no age (except perhaps the golden [82]age of Greece)which has a more convincing proof to offer of the transcendent genius of itsgreat men. Of the three problems, that of the infinitesimal was solved byWeierstrass; the solution of the other two was begun by Dedekind, anddefinitively accomplished by Cantor. The infinitesimal played formerly a great part in mathematics. It wasintroduced by the Greeks, who regarded a circle as differing infinitesimallyfrom a polygon with a very large number of very small equal sides. It graduallygrew in importance, until, when Leibniz invented the Infinitesimal Calculus, itseemed to become the fundamental notion of all higher mathematics. Carlyletells, in his Frederick the Great, how Leibniz used to discourse to QueenSophia Charlotte of Prussia concerning the infinitely little, and how she wouldreply that on that subject she needed no instruction—the behaviour of courtiershad made her thoroughly familiar with it. But philosophers andmathematicians—who for the most part had less acquaintance with courts—continued to discuss this topic, though without making any advance. TheCalculus required continuity, and continuity was supposed to require theinfinitely little; but nobody could discover what the infinitely little might be. Itwas plainly not quite zero, because a sufficiently large number ofinfinitesimals, added together, were seen to make up a finite whole. But nobodycould point out any fraction which was not zero, and yet not finite. Thus therewas a deadlock. But at last Weierstrass discovered that the infinitesimal wasnot needed at all, and that everything could be accomplished without it. Thusthere was no longer any need to suppose that there was such a thing.Nowadays, therefore, mathematicians are more dignified than Leibniz: insteadof talking about the infinitely small, they talk about the infinitely great—asubject [83]which, however appropriate to monarchs, seems, unfortunately, tointerest them even less than the infinitely little interested the monarchs towhom Leibniz discoursed. The banishment of the infinitesimal has all sorts of odd consequences, towhich one has to become gradually accustomed. For example, there is no suchthing as the next moment. The interval between one moment and the nextwould have to be infinitesimal, since, if we take two moments with a finiteinterval between them, there are always other moments in the interval. Thus ifthere are to be no infinitesimals, no two moments are quite consecutive, butthere are always other moments between any two. Hence there must be aninfinite number of moments between any two; because if there were a finitenumber one would be nearest the first of the two moments, and therefore nextto it. This might be thought to be a difficulty; but, as a matter of fact, it is herethat the philosophy of the infinite comes in, and makes all straight. The same sort of thing happens in space. If any piece of matter be cut in two,and then each part be halved, and so on, the bits will become smaller andsmaller, and can theoretically be made as small as we please. However smallthey may be, they can still be cut up and made smaller still. But they willalways have some finite size, however small they may be. We never reach theinfinitesimal in this way, and no finite number of divisions will bring us topoints. Nevertheless there are points, only these are not to be reached bysuccessive divisions. Here again, the philosophy of the infinite shows us howthis is possible, and why points are not infinitesimal lengths. As regards motion and change, we get similarly curious results. People usedto think that when a thing changes, it must be in a state of change, and thatwhen a thing[84]moves, it is in a state of motion. This is now known to be amistake. When a body moves, all that can be said is that it is in one place at onetime and in another at another. We must not say that it will be in aneighbouring place at the next instant, since there is no next instant.Philosophers often tell us that when a body is in motion, it changes its positionwithin the instant. To this view Zeno long ago made the fatal retort that everybody always is where it is; but a retort so simple and brief was not of the kindto which philosophers are accustomed to give weight, and they have continueddown to our own day to repeat the same phrases which roused the Eleatic'sdestructive ardour. It was only recently that it became possible to explainmotion in detail in accordance with Zeno's platitude, and in opposition to thephilosopher's paradox. We may now at last indulge the comfortable belief that abody in motion is just as truly where it is as a body at rest. Motion consistsmerely in the fact that bodies are sometimes in one place and sometimes inanother, and that they are at intermediate places at intermediate times. Onlythose who have waded through the quagmire of philosophic speculation on thissubject can realise what a liberation from antique prejudices is involved in thissimple and straightforward commonplace. The philosophy of the infinitesimal, as we have just seen, is mainly negative.People used to believe in it, and now they have found out their mistake. Thephilosophy of the infinite, on the other hand, is wholly positive. It was formerlysupposed that infinite numbers, and the mathematical infinite generally, wereself-contradictory. But as it was obvious that there were infinities—forexample, the number of numbers—the contradictions of infinity seemedunavoidable, and philosophy seemed to [85]have wandered into a "cul-de-sac."This difficulty led to Kant's antinomies, and hence, more or less indirectly, tomuch of Hegel's dialectic method. Almost all current philosophy is upset by thefact (of which very few philosophers are as yet aware) that all the ancient andrespectable contradictions in the notion of the infinite have been once for alldisposed of. The method by which this has been done is most interesting andinstructive. In the first place, though people had talked glibly about infinityever since the beginnings of Greek thought, nobody had ever thought of asking,What is infinity? If any philosopher had been asked for a definition of infinity,he might have produced some unintelligible rigmarole, but he would certainlynot have been able to give a definition that had any meaning at all. Twentyyears ago, roughly speaking, Dedekind and Cantor asked this question, and,what is more remarkable, they answered it. They found, that is to say, aperfectly precise definition of an infinite number or an infinite collection ofthings. This was the first and perhaps the greatest step. It then remained toexamine the supposed contradictions in this notion. Here Cantor proceeded inthe only proper way. He took pairs of contradictory propositions, in which bothsides of the contradiction would be usually regarded as demonstrable, and hestrictly examined the supposed proofs. He found that all proofs adverse toinfinity involved a certain principle, at first sight obviously true, butdestructive, in its consequences, of almost all mathematics. The proofsfavourable to infinity, on the other hand, involved no principle that had evilconsequences. It thus appeared that common sense had allowed itself to betaken in by a specious maxim, and that, when once this maxim was rejected, allwent well. The maxim in question is, that if one collection is part [86]of another, the onewhich is a part has fewer terms than the one of which it is a part. This maxim istrue of finite numbers. For example, Englishmen are only some amongEuropeans, and there are fewer Englishmen than Europeans. But when wecome to infinite numbers, this is no longer true. This breakdown of the maximgives us the precise definition of infinity. A collection of terms is infinite whenit contains as parts other collections which have just as many terms as it has. Ifyou can take away some of the terms of a collection, without diminishing thenumber of terms, then there are an infinite number of terms in the collection.For example, there are just as many even numbers as there are numbersaltogether, since every number can be doubled. This may be seen by puttingodd and even numbers together in one row, and even numbers alone in a rowbelow:—1, 2, 3, 4, 5, ad infinitum.2, 4, 6, 8, 10, ad infinitum.There are obviously just as many numbers in the row below as in the rowabove, because there is one below for each one above. This property, whichwas formerly thought to be a contradiction, is now transformed into a harmlessdefinition of infinity, and shows, in the above case, that the number of finitenumbers is infinite. But the uninitiated may wonder how it is possible to deal with a numberwhich cannot be counted. It is impossible to count up all the numbers, one byone, because, however many we may count, there are always more to follow.The fact is that counting is a very vulgar and elementary way of finding outhow many terms there are in a collection. And in any case, counting gives uswhat mathematicians call the ordinal number of our terms; that is to say, itarranges our terms in an order or [87]series, and its result tells us what type ofseries results from this arrangement. In other words, it is impossible to countthings without counting some first and others afterwards, so that countingalways has to do with order. Now when there are only a finite number of terms,we can count them in any order we like; but when there are an infinite number,what corresponds to counting will give us quite different results according tothe way in which we carry out the operation. Thus the ordinal number, whichresults from what, in a general sense may be called counting, depends not onlyupon how many terms we have, but also (where the number of terms is infinite)upon the way in which the terms are arranged. The fundamental infinite numbers are not ordinal, but are what iscalled cardinal. They are not obtained by putting our terms in order andcounting them, but by a different method, which tells us, to begin with, whethertwo collections have the same number of terms, or, if not, which is thegreater.[14] It does not tell us, in the way in which counting does, what numberof terms a collection has; but if we define a number as the number of terms insuch and such a collection, then this method enables us to discover whethersome other collection that may be mentioned has more or fewer terms. Anillustration will show how this is done. If there existed some country in which,for one reason or another, it was impossible to take a census, but in which itwas known that every man had a wife and every woman a husband, then(provided polygamy was not a national institution) we should know, withoutcounting, that there were exactly as many men as there were women in thatcountry, neither more nor[88]less. This method can be applied generally. If thereis some relation which, like marriage, connects the things in one collection eachwith one of the things in another collection, and vice versa, then the twocollections have the same number of terms. This was the way in which wefound that there are as many even numbers as there are numbers. Every numbercan be doubled, and every even number can be halved, and each process givesjust one number corresponding to the one that is doubled or halved. And in thisway we can find any number of collections each of which has just as manyterms as there are finite numbers. If every term of a collection can be hookedon to a number, and all the finite numbers are used once, and only once, in theprocess, then our collection must have just as many terms as there are finitenumbers. This is the general method by which the numbers of infinitecollections are defined. But it must not be supposed that all infinite numbers are equal. On thecontrary, there are infinitely more infinite numbers than finite ones. There aremore ways of arranging the finite numbers in different types of series than thereare finite numbers. There are probably more points in space and more momentsin time than there are finite numbers. There are exactly as many fractions aswhole numbers, although there are an infinite number of fractions between anytwo whole numbers. But there are more irrational numbers than there are wholenumbers or fractions. There are probably exactly as many points in space asthere are irrational numbers, and exactly as many points on a line a millionth ofan inch long as in the whole of infinite space. There is a greatest of all infinitenumbers, which is the number of things altogether, of every sort and kind. It isobvious that there cannot be a greater number than this, because, [89]ifeverything has been taken, there is nothing left to add. Cantor has a proof thatthere is no greatest number, and if this proof were valid, the contradictions ofinfinity would reappear in a sublimated form. But in this one point, the masterhas been guilty of a very subtle fallacy, which I hope to explain in some futurework.[15] We can now understand why Zeno believed that Achilles cannot overtake thetortoise and why as a matter of fact he can overtake it. We shall see that all thepeople who disagreed with Zeno had no right to do so, because they allaccepted premises from which his conclusion followed. The argument is this:Let Achilles and the tortoise start along a road at the same time, the tortoise (asis only fair) being allowed a handicap. Let Achilles go twice as fast as thetortoise, or ten times or a hundred times as fast. Then he will never reach thetortoise. For at every moment the tortoise is somewhere and Achilles issomewhere; and neither is ever twice in the same place while the race is goingon. Thus the tortoise goes to just as many places as Achilles does, because eachis in one place at one moment, and in another at any other moment. But ifAchilles were to catch up with the tortoise, the places where the tortoise wouldhave been would be only part of the places where Achilles would have been.Here, we must suppose, Zeno appealed to the maxim that the whole has moreterms than the part.[16] Thus if Achilles were [90]to overtake the tortoise, hewould have been in more places than the tortoise; but we saw that he must, inany period, be in exactly as many places as the tortoise. Hence we infer that hecan never catch the tortoise. This argument is strictly correct, if we allow theaxiom that the whole has more terms than the part. As the conclusion is absurd,the axiom must be rejected, and then all goes well. But there is no good word tobe said for the philosophers of the past two thousand years and more, who haveall allowed the axiom and denied the conclusion. The retention of this axiom leads to absolute contradictions, while itsrejection leads only to oddities. Some of these oddities, it must be confessed,are very odd. One of them, which I call the paradox of Tristram Shandy, is theconverse of the Achilles, and shows that the tortoise, if you give him time, willgo just as far as Achilles. Tristram Shandy, as we know, employed two years inchronicling the first two days of his life, and lamented that, at this rate, materialwould accumulate faster than he could deal with it, so that, as years went by, hewould be farther and farther from the end of his history. Now I maintain that, ifhe had lived for ever, and had not wearied of his task, then, even if his life hadcontinued as event fully as it began, no part of his biography would haveremained unwritten. For consider: the hundredth day will be described in thehundredth year, the thousandth in the thousandth year, and so on. Whatever daywe may choose as so far on that he cannot hope to reach it, that day will bedescribed in the corresponding year. Thus any day that may be mentioned willbe written up sooner or later, and therefore no part of the biography will remainpermanently unwritten. This paradoxical but perfectly true proposition dependsupon the fact [91]that the number of days in all time is no greater than thenumber of years. Thus on the subject of infinity it is impossible to avoid conclusions which atfirst sight appear paradoxical, and this is the reason why so many philosophershave supposed that there were inherent contradictions in the infinite. But a littlepractice enables one to grasp the true principles of Cantor's doctrine, and toacquire new and better instincts as to the true and the false. The oddities thenbecome no odder than the people at the antipodes, who used to be thoughtimpossible because they would find it so inconvenient to stand on their heads. The solution of the problems concerning infinity has enabled Cantor to solvealso the problems of continuity. Of this, as of infinity, he has given a perfectlyprecise definition, and has shown that there are no contradictions in the notionso defined. But this subject is so technical that it is impossible to give anyaccount of it here. The notion of continuity depends upon that of order, since continuity ismerely a particular type of order. Mathematics has, in modern times, broughtorder into greater and greater prominence. In former days, it was supposed (andphilosophers are still apt to suppose) that quantity was the fundamental notionof mathematics. But nowadays, quantity is banished altogether, except fromone little corner of Geometry, while order more and more reigns supreme. Theinvestigation of different kinds of series and their relations is now a very largepart of mathematics, and it has been found that this investigation can beconducted without any reference to quantity, and, for the most part, without anyreference to number. All types of series are capable of formal definition, andtheir properties can be deduced from the principles of symbolic logic by meansof the Algebra of Relatives. [92]The notion of a limit, which is fundamental inthe greater part of higher mathematics, used to be defined by means of quantity,as a term to which the terms of some series approximate as nearly as we please.But nowadays the limit is defined quite differently, and the series which itlimits may not approximate to it at all. This improvement also is due to Cantor,and it is one which has revolutionised mathematics. Only order is now relevantto limits. Thus, for instance, the smallest of the infinite integers is the limit ofthe finite integers, though all finite integers are at an infinite distance from it.The study of different types of series is a general subject of which the study ofordinal numbers (mentioned above) is a special and very interesting branch.But the unavoidable technicalities of this subject render it impossible to explainto any but professed mathematicians. Geometry, like Arithmetic, has been subsumed, in recent times, under thegeneral study of order. It was formerly supposed that Geometry was the studyof the nature of the space in which we live, and accordingly it was urged, bythose who held that what exists can only be known empirically, that Geometryshould really be regarded as belonging to applied mathematics. But it hasgradually appeared, by the increase of non-Euclidean systems, that Geometrythrows no more light upon the nature of space than Arithmetic throws upon thepopulation of the United States. Geometry is a whole collection of deductivesciences based on a corresponding collection of sets of axioms. One set ofaxioms is Euclid's; other equally good sets of axioms lead to other results.Whether Euclid's axioms are true, is a question as to which the puremathematician is indifferent; and, what is more, it is a question which it istheoretically impossible to answer with certainty in the affirmative. Itmight [93]possibly be shown, by very careful measurements, that Euclid'saxioms are false; but no measurements could ever assure us (owing to theerrors of observation) that they are exactly true. Thus the geometer leaves to theman of science to decide, as best he may, what axioms are most nearly true inthe actual world. The geometer takes any set of axioms that seem interesting,and deduces their consequences. What defines Geometry, in this sense, is thatthe axioms must give rise to a series of more than one dimension. And it is thusthat Geometry becomes a department in the study of order. In Geometry, as in other parts of mathematics, Peano and his disciples havedone work of the very greatest merit as regards principles. Formerly, it washeld by philosophers and mathematicians alike that the proofs in Geometrydepended on the figure; nowadays, this is known to be false. In the best booksthere are no figures at all. The reasoning proceeds by the strict rules of formallogic from a set of axioms laid down to begin with. If a figure is used, all sortsof things seem obviously to follow, which no formal reasoning can prove fromthe explicit axioms, and which, as a matter of fact, are only accepted becausethey are obvious. By banishing the figure, it becomes possible todiscover all the axioms that are needed; and in this way all sorts of possibilities,which would have otherwise remained undetected, are brought to light. One great advance, from the point of view of correctness, has been made byintroducing points as they are required, and not starting, as was formerly done,by assuming the whole of space. This method is due partly to Peano, partly toanother Italian named Fano. To those unaccustomed to it, it has an air ofsomewhat wilful pedantry. In this way, we begin with the following [94]axioms:(1) There is a class of entities called points. (2) There is at least one point. (3)If a be a point, there is at least one other point besides a. Then we bring in thestraight line joining two points, and begin again with (4), namely, on thestraight line joining a and b, there is at least one other point besides a and b. (5)There is at least one point not on the line ab. And so we go on, till we have themeans of obtaining as many points as we require. But the word space, as Peanohumorously remarks, is one for which Geometry has no use at all. The rigid methods employed by modern geometers have deposed Euclidfrom his pinnacle of correctness. It was thought, until recent times, that, as SirHenry Savile remarked in 1621, there were only two blemishes in Euclid, thetheory of parallels and the theory of proportion. It is now known that these arealmost the only points in which Euclid is free from blemish. Countless errorsare involved in his first eight propositions. That is to say, not only is it doubtfulwhether his axioms are true, which is a comparatively trivial matter, but it iscertain that his propositions do not follow from the axioms which heenunciates. A vastly greater number of axioms, which Euclid unconsciouslyemploys, are required for the proof of his propositions. Even in the firstproposition of all, where he constructs an equilateral triangle on a given base,he uses two circles which are assumed to intersect. But no explicit axiomassures us that they do so, and in some kinds of spaces they do not alwaysintersect. It is quite doubtful whether our space belongs to one of these kinds ornot. Thus Euclid fails entirely to prove his point in the very first proposition. Ashe is certainly not an easy author, and is terribly long-winded, he has no longerany but an historical interest. Under these circumstances, it is nothing less thana [95]scandal that he should still be taught to boys in England.[17] A book shouldhave either intelligibility or correctness; to combine the two is impossible, butto lack both is to be unworthy of such a place as Euclid has occupied ineducation. The most remarkable result of modern methods in mathematics is theimportance of symbolic logic and of rigid formalism. Mathematicians, underthe influence of Weierstrass, have shown in modern times a care for accuracy,and an aversion to slipshod reasoning, such as had not been known among thempreviously since the time of the Greeks. The great inventions of the seventeenthcentury—Analytical Geometry and the Infinitesimal Calculus—were so fruitfulin new results that mathematicians had neither time nor inclination to examinetheir foundations. Philosophers, who should have taken up the task, had toolittle mathematical ability to invent the new branches of mathematics whichhave now been found necessary for any adequate discussion. Thusmathematicians were only awakened from their "dogmatic slumbers" whenWeierstrass and his followers showed that many of their most cherishedpropositions are in general false. Macaulay, contrasting the certainty ofmathematics with the uncertainty of philosophy, asks who ever heard of areaction against Taylor's theorem? If he had lived now, he himself might haveheard of such a reaction, for this is precisely one of the theorems which moderninvestigations have overthrown. Such rude shocks to mathematical faith haveproduced that love of formalism which appears, to those who are ignorant of itsmotive, to be mere outrageous pedantry. [96]The proof that all pure mathematics, including Geometry, is nothing butformal logic, is a fatal blow to the Kantian philosophy. Kant, rightly perceivingthat Euclid's propositions could not be deduced from Euclid's axioms withoutthe help of the figures, invented a theory of knowledge to account for this fact;and it accounted so successfully that, when the fact is shown to be a meredefect in Euclid, and not a result of the nature of geometrical reasoning, Kant'stheory also has to be abandoned. The whole doctrine of a priori intuitions, bywhich Kant explained the possibility of pure mathematics, is whollyinapplicable to mathematics in its present form. The Aristotelian doctrines ofthe schoolmen come nearer in spirit to the doctrines which modernmathematics inspire; but the schoolmen were hampered by the fact that theirformal logic was very defective, and that the philosophical logic based upon thesyllogism showed a corresponding narrowness. What is now required is to givethe greatest possible development to mathematical logic, to allow to the full theimportance of relations, and then to found upon this secure basis a newphilosophical logic, which may hope to borrow some of the exactitude andcertainty of its mathematical foundation. If this can be successfullyaccomplished, there is every reason to hope that the near future will be as greatan epoch in pure philosophy as the immediate past has been in the principles ofmathematics. Great triumphs inspire great hopes; and pure thought mayachieve, within our generation, such results as will place our time, in thisrespect, on a level with the greatest age of Greece.[18] FOOTNOTES:

[11]This subject is due in the main to Mr. C.S. Peirce.

[12]I ought to have added Frege, but his writings were unknown to me when this article was written.[Note added in 1917.][13]Professor of Mathematics in the University of Berlin. He died in 1897.[14][Note added in 1917.] Although some infinite numbers are greater than some others, it cannot beproved that of any two infinite numbers one must be the greater.[15]Cantor was not guilty of a fallacy on this point. His proof that there is no greatest number is valid.The solution of the puzzle is complicated and depends upon the theory of types, which is explainedin Principia Mathematica, Vol. I (Camb. Univ. Press, 1910). [Note added in 1917.][16]This must not be regarded as a historically correct account of what Zeno actually had in mind. It isa new argument for his conclusion, not the argument which influenced him. On this point, see e.g.C.D. Broad, "Note on Achilles and the Tortoise," Mind, N.S., Vol. XXII, pp. 318-19. Much valuablework on the interpretation of Zeno has been done since this article was written. [Note added in 1917.][17]Since the above was written, he has ceased to be used as a textbook. But I fear many of the booksnow used are so bad that the change is no great improvement. [Note added in 1917.][18]The greatest age of Greece was brought to an end by the Peloponnesian War. [Note added in1917.]

[97]

VIToC

ON SCIENTIFIC METHOD IN PHILOSOPHY

When we try to ascertain the motives which have led men to the investigationof philosophical questions, we find that, broadly speaking, they can be dividedinto two groups, often antagonistic, and leading to very divergent systems.These two groups of motives are, on the one hand, those derived from religionand ethics, and, on the other hand, those derived from science. Plato, Spinoza,and Hegel may be taken as typical of the philosophers whose interests aremainly religious and ethical, while Leibniz, Locke, and Hume may be taken asrepresentatives of the scientific wing. In Aristotle, Descartes, Berkeley, andKant we find both groups of motives strongly present. Herbert Spencer, in whose honour we are assembled to-day, would naturallybe classed among scientific philosophers: it was mainly from science that hedrew his data, his formulation of problems, and his conception of method. Buthis strong religious sense is obvious in much of his writing, and his ethical pre-occupations are what make him value the conception of evolution—thatconception in which, as a whole generation has believed, science and moralsare to be united in fruitful and indissoluble marriage. It is my belief that the ethical and religious motives [98]in spite of thesplendidly imaginative systems to which they have given rise, have been on thewhole a hindrance to the progress of philosophy, and ought now to beconsciously thrust aside by those who wish to discover philosophical truth.Science, originally, was entangled in similar motives, and was thereby hinderedin its advances. It is, I maintain, from science, rather than from ethics andreligion, that philosophy should draw its inspiration. But there are two different ways in which a philosophy may seek to baseitself upon science. It may emphasise the most general results of science, andseek to give even greater generality and unity to these results. Or it may studythe methods of science, and seek to apply these methods, with the necessaryadaptations, to its own peculiar province. Much philosophy inspired by sciencehas gone astray through preoccupation with the results momentarily supposedto have been achieved. It is not results, but methods that can be transferred withprofit from the sphere of the special sciences to the sphere of philosophy. WhatI wish to bring to your notice is the possibility and importance of applying tophilosophical problems certain broad principles of method which have beenfound successful in the study of scientific questions. The opposition between a philosophy guided by scientific method and aphilosophy dominated by religious and ethical ideas may be illustrated by twonotions which are very prevalent in the works of philosophers, namely thenotion of the universe, and the notion of good and evil. A philosopher isexpected to tell us something about the nature of the universe as a whole, and togive grounds for either optimism or pessimism. Both these expectations seemto me mistaken. I believe the conception of "the universe" to be, as itsetymology indicates, a [99]mere relic of pre-Copernican astronomy: and Ibelieve the question of optimism and pessimism to be one which thephilosopher will regard as outside his scope, except, possibly, to the extent ofmaintaining that it is insoluble. In the days before Copernicus, the conception of the "universe" wasdefensible on scientific grounds: the diurnal revolution of the heavenly bodiesbound them together as all parts of one system, of which the earth was thecentre. Round this apparent scientific fact, many human desires rallied: thewish to believe Man important in the scheme of things, the theoretical desirefor a comprehensive understanding of the Whole, the hope that the course ofnature might be guided by some sympathy with our wishes. In this way, anethically inspired system of metaphysics grew up, whose anthropocentrism wasapparently warranted by the geocentrism of astronomy. When Copernicusswept away the astronomical basis of this system of thought, it had grown sofamiliar, and had associated itself so intimately with men's aspirations, that itsurvived with scarcely diminished force—survived even Kant's "Copernicanrevolution," and is still now the unconscious premiss of most metaphysicalsystems. The oneness of the world is an almost undiscussed postulate of mostmetaphysics. "Reality is not merely one and self-consistent, but is a system ofreciprocally determinate parts"[19]—such a statement would pass almostunnoticed as a mere truism. Yet I believe that it embodies a failure to effectthoroughly the "Copernican revolution," and that the apparent oneness of theworld is merely the oneness of what is seen by a single spectator orapprehended by a single mind. The Critical Philosophy, although it intended toemphasise the subjective element [100]in many apparent characteristics of theworld, yet, by regarding the world in itself as unknowable, so concentratedattention upon the subjective representation that its subjectivity was soonforgotten. Having recognised the categories as the work of the mind, it wasparalysed by its own recognition, and abandoned in despair the attempt to undothe work of subjective falsification. In part, no doubt, its despair was wellfounded, but not, I think, in any absolute or ultimate sense. Still less was it aground for rejoicing, or for supposing that the nescience to which it ought tohave given rise could be legitimately exchanged for a metaphysical dogmatism.

As regards our present question, namely, the question of the unity of theworld, the right method, as I think, has been indicated by WilliamJames.[20] "Let us now turn our backs upon ineffable or unintelligible ways ofaccounting for the world's oneness, and inquire whether, instead of being aprinciple, the 'oneness' affirmed may not merely be a name like 'substance'descriptive of the fact that certain specific and verifiable connections are foundamong the parts of the experiential flux.... We can easily conceive of things thatshall have no connection whatever with each other. We may assume them toinhabit different times and spaces, as the dreams of different persons do evennow. They may be so unlike and incommensurable, and so inert towards oneanother, as never to jostle or interfere. Even now there may actually be wholeuniverses so disparate from ours that we who know ours have no means ofperceiving that they exist. We conceive their diversity, however; and bythat [101]fact the whole lot of them form what is known in logic as 'a universe ofdiscourse.' To form a universe of discourse argues, as this example shows, nofurther kind of connexion. The importance attached by certain monistic writersto the fact that any chaos may become a universe by merely being named, is tome incomprehensible." We are thus left with two kinds of unity in theexperienced world; the one what we may call the epistemological unity, duemerely to the fact that my experienced world is what oneexperience selectsfrom the sum total of existence: the other that tentative and partial unityexhibited in the prevalence of scientific laws in those portions of the worldwhich science has hitherto mastered. Now a generalisation based upon either ofthese kinds of unity would be fallacious. That the things which we experiencehave the common property of being experienced by us is a truism from whichobviously nothing of importance can be deducible: it is clearly fallacious todraw from the fact that whatever we experience is experienced the conclusionthat therefore everything must be experienced. The generalisation of the secondkind of unity, namely, that derived from scientific laws, would be equallyfallacious, though the fallacy is a trifle less elementary. In order to explain it letus consider for a moment what is called the reign of law. People often speak asthough it were a remarkable fact that the physical world is subject to invariablelaws. In fact, however, it is not easy to see how such a world could fail to obeygeneral laws. Taking any arbitrary set of points in space, there is a function ofthe time corresponding to these points, i.e. expressing the motion of a particlewhich traverses these points: this function may be regarded as a general law towhich the behaviour of such a particle is subject. Taking all such functionsfor [102]all the particles in the universe, there will be theoretically some oneformula embracing them all, and this formula may be regarded as the single andsupreme law of the spatio-temporal world. Thus what is surprising in physics isnot the existence of general laws, but their extreme simplicity. It is not theuniformity of nature that should surprise us, for, by sufficient analyticingenuity, any conceivable course of nature might be shown to exhibituniformity. What should surprise us is the fact that the uniformity is simpleenough for us to be able to discover it. But it is just this characteristic ofsimplicity in the laws of nature hitherto discovered which it would be fallaciousto generalise, for it is obvious that simplicity has been a part cause of theirdiscovery, and can, therefore, give no ground for the supposition that otherundiscovered laws are equally simple. The fallacies to which these two kinds of unity have given rise suggest acaution as regards all use in philosophy of general results that science issupposed to have achieved. In the first place, in generalising these resultsbeyond past experience, it is necessary to examine very carefully whether thereis not some reason making it more probable that these results should hold of allthat has been experienced than that they should hold of things universally. Thesum total of what is experienced by mankind is a selection from the sum totalof what exists, and any general character exhibited by this selection may be dueto the manner of selecting rather than to the general character of that fromwhich experience selects. In the second place, the most general results ofscience are the least certain and the most liable to be upset by subsequentresearch. In utilizing these results as the basis of a philosophy, we sacrifice themost valuable and remarkable characteristic of scientific method, [103]namely,that, although almost everything in science is found sooner or later to requiresome correction, yet this correction is almost always such as to leaveuntouched, or only slightly modified, the greater part of the results which havebeen deduced from the premiss subsequently discovered to be faulty. Theprudent man of science acquires a certain instinct as to the kind of uses whichmay be made of present scientific beliefs without incurring the danger ofcomplete and utter refutation from the modifications likely to be introduced bysubsequent discoveries. Unfortunately the use of scientific generalisations of asweeping kind as the basis of philosophy is just that kind of use which aninstinct of scientific caution would avoid, since, as a rule, it would only lead totrue results if the generalisation upon which it is based stood in no need ofcorrection. We may illustrate these general considerations by means of two examples,namely, the conservation of energy and the principle of evolution. (1) Let us begin with the conservation of energy, or, as Herbert Spencer usedto call it, the persistence of force. He says:[21] "Before taking a first step in the rational interpretation of Evolution, it isneedful to recognise, not only the facts that Matter is indestructible and Motioncontinuous, but also the fact that Force persists. An attempt to assignthe causes of Evolution would manifestly be absurd if that agency to which themetamorphosis in general and in detail is due, could either come into existenceor cease to exist. The succession of phenomena would in such case bealtogether arbitrary, and deductive Science impossible." [104]This paragraph illustrates the kind of way in which the philosopher istempted to give an air of absoluteness and necessity to empiricalgeneralisations, of which only the approximate truth in the regions hithertoinvestigated can be guaranteed by the unaided methods of science. It is veryoften said that the persistence of something or other is a necessarypresupposition of all scientific investigation, and this presupposition is thenthought to be exemplified in some quantity which physics declares to beconstant. There are here, as it seems to me, three distinct errors. First, thedetailed scientific investigation of nature does not presuppose any such generallaws as its results are found to verify. Apart from particular observations,science need presuppose nothing except the general principles of logic, andthese principles are not laws of nature, for they are merely hypothetical, andapply not only to the actual world but to whatever is possible. The second errorconsists in the identification of a constant quantity with a persistent entity.Energy is a certain function of a physical system, but is not a thing or substancepersisting throughout the changes of the system. The same is true of mass, inspite of the fact that mass has often been defined as quantity of matter. Thewhole conception, of quantity, involving, as it does, numerical measurementbased largely upon conventions, is far more artificial, far more an embodimentof mathematical convenience, than is commonly believed by those whophilosophise on physics. Thus even if (which I cannot for a moment admit) thepersistence of some entity were among the necessary postulates of science, itwould be a sheer error to infer from this the constancy of any physical quantity,or the a priori necessity of any such constancy which may be empiricallydiscovered. In the third place, it [105]has become more and more evident withthe progress of physics that large generalisations, such as the conservation ofenergy or mass, are far from certain and are very likely only approximate.Mass, which used to be regarded as the most indubitable of physical quantities,is now generally believed to vary according to velocity, and to be, in fact, avector quantity which at a given moment is different in different directions. Thedetailed conclusions deduced from the supposed constancy of mass for suchmotions as used to be studied in physics will remain very nearly exact, andtherefore over the field of the older investigations very little modification of theolder results is required. But as soon as such a principle as the conservation ofmass or of energy is erected into a universal a priori law, the slightest failure inabsolute exactness is fatal, and the whole philosophic structure raised upon thisfoundation is necessarily ruined. The prudent philosopher, therefore, though hemay with advantage study the methods of physics, will be very chary of basinganything upon what happen at the moment to be the most general resultsapparently obtained by those methods. (2) The philosophy of evolution, which was to be our second example,illustrates the same tendency to hasty generalisation, and also another sort,namely, the undue preoccupation with ethical notions. There are two kinds ofevolutionist philosophy, of which both Hegel and Spencer represent the olderand less radical kind, while Pragmatism and Bergson represent the moremodern and revolutionary variety. But both these sorts of evolutionism have incommon the emphasis on progress, that is, upon a continual change from theworse to the better, or from the simpler to the more complex. It [106]would beunfair to attribute to Hegel any scientific motive or foundation, but all the otherevolutionists, including Hegel's modern disciples, have derived their impetusvery largely from the history of biological development. To a philosophy whichderives a law of universal progress from this history there are two objections.First, that this history itself is concerned with a very small selection of factsconfined to an infinitesimal fragment of space and time, and even on scientificgrounds probably not an average sample of events in the world at large. For weknow that decay as well as growth is a normal occurrence in the world. Anextra-terrestrial philosopher, who had watched a single youth up to the age oftwenty-one and had never come across any other human being, might concludethat it is the nature of human beings to grow continually taller and wiser in anindefinite progress towards perfection; and this generalisation would be just aswell founded as the generalisation which evolutionists base upon the previoushistory of this planet. Apart, however, from this scientific objection toevolutionism, there is another, derived from the undue admixture of ethicalnotions in the very idea of progress from which evolutionism derives its charm.Organic life, we are told, has developed gradually from the protozoon to thephilosopher, and this development, we are assured, is indubitably an advance.Unfortunately it is the philosopher, not the protozoon, who gives us thisassurance, and we can have no security that the impartial outsider would agreewith the philosopher's self-complacent assumption. This point has beenillustrated by the philosopher Chuang Tzŭ in the following instructiveanecdote: "The Grand Augur, in his ceremonial robes, approached the shambles [107] andthus addressed the pigs: 'How can you object to die? I shall fatten you for threemonths. I shall discipline myself for ten days and fast for three. I shall strewfine grass, and place you bodily upon a carved sacrificial dish. Does not thissatisfy you?' Then, speaking from the pigs' point of view, he continued: 'It is better,perhaps, after all, to live on bran and escape the shambles....' 'But then,' added he, speaking from his own point of view,' to enjoy honourwhen alive one would readily die on a war-shield or in the headsman's basket.' So he rejected the pigs' point of view and adopted his own point of view. Inwhat sense, then, was he different from the pigs?" I much fear that the evolutionists too often resemble the Grand Augur and thepigs. The ethical element which has been prominent in many of the most famoussystems of philosophy is, in my opinion, one of the most serious obstacles tothe victory of scientific method in the investigation of philosophical questions.Human ethical notions, as Chuang Tzŭ perceived, are essentiallyanthropocentric, and involve, when used in metaphysics, an attempt, howeverveiled, to legislate for the universe on the basis of the present desires of men. Inthis way they interfere with that receptivity to fact which is the essence of thescientific attitude towards the world. To regard ethical notions as a key to theunderstanding of the world is essentially pre-Copernican. It is to make man,with the hopes and ideals which he happens to have at the present moment, thecentre of the universe and the interpreter of its supposed aims and purposes.Ethical metaphysics is fundamentally an attempt, however disguised,to [108]give legislative force to our own wishes. This may, of course, bequestioned, but I think that it is confirmed by a consideration of the way inwhich ethical notions arise. Ethics is essentially a product of the gregariousinstinct, that is to say, of the instinct to co-operate with those who are to formour own group against those who belong to other groups. Those who belong toour own group are good; those who belong to hostile groups are wicked. Theends which are pursued by our own group are desirable ends, the ends pursuedby hostile groups are nefarious. The subjectivity of this situation is not apparentto the gregarious animal, which feels that the general principles of justice areon the side of its own herd. When the animal has arrived at the dignity of themetaphysician, it invents ethics as the embodiment of its belief in the justice ofits own herd. So the Grand Augur invokes ethics as the justification of Augursin their conflicts with pigs. But, it may be said, this view of ethics takes noaccount of such truly ethical notions as that of self-sacrifice. This, however,would be a mistake. The success of gregarious animals in the struggle forexistence depends upon co-operation within the herd, and co-operation requiressacrifice, to some extent, of what would otherwise be the interest of theindividual. Hence arises a conflict of desires and instincts, since both self-preservation and the preservation of the herd are biological ends to theindividual. Ethics is in origin the art of recommending to others the sacrificesrequired for co-operation with oneself. Hence, by reflexion, it comes, throughthe operation of social justice, to recommend sacrifices by oneself, but allethics, however refined, remains more or less subjective. Even vegetarians donot hesitate, for example, to save the life of a man in a fever, although in doingso they destroy the lives of many millions of m[109]icrobes. The view of theworld taken by the philosophy derived from ethical notions is thus neverimpartial and therefore never fully scientific. As compared with science, it failsto achieve the imaginative liberation from self which is necessary to suchunderstanding of the world as man can hope to achieve, and the philosophywhich it inspires is always more or less parochial, more or less infected withthe prejudices of a time and a place. I do not deny the importance or value, within its own sphere, of the kind ofphilosophy which is inspired by ethical notions. The ethical work of Spinoza,for example, appears to me of the very highest significance, but what isvaluable in such work is not any metaphysical theory as to the nature of theworld to which it may give rise, nor indeed anything which can be proved ordisproved by argument. What is valuable is the indication of some new way offeeling towards life and the world, some way of feeling by which our ownexistence can acquire more of the characteristics which we must deeply desire.The value of such work, however immeasurable it is, belongs with practice andnot with theory. Such theoretic importance as it may possess is only in relationto human nature, not in relation to the world at large. The scientific philosophy,therefore, which aims only at understanding the world and not directly at anyother improvement of human life, cannot take account of ethical notionswithout being turned aside from that submission to fact which is the essence ofthe scientific temper.[110]

II

If the notion of the universe and the notion of good and evil are extrudedfrom scientific philosophy, it may be asked what specific problems remain forthe philosopher as opposed to the man of science? It would be difficult to givea precise answer to this question, but certain characteristics may be noted asdistinguishing the province of philosophy from that of the special sciences. In the first place a philosophical proposition must be general. It must not dealspecially with things on the surface of the earth, or with the solar system, orwith any other portion of space and time. It is this need of generality which hasled to the belief that philosophy deals with the universe as a whole. I do notbelieve that this belief is justified, but I do believe that a philosophicalproposition must be applicable to everything that exists or may exist. It mightbe supposed that this admission would be scarcely distinguishable from theview which I wish to reject. This, however, would be an error, and an importantone. The traditional view would make the universe itself the subject of variouspredicates which could not be applied to any particular thing in the universe,and the ascription of such peculiar predicates to the universe would be thespecial business of philosophy. I maintain, on the contrary, that there are nopropositions of which the "universe" is the subject; in other words, that there isno such thing as the "universe." What I do maintain is that there are generalpropositions which may be asserted of each individual thing, such as thepropositions of logic. This does not involve that all the things there are form awhole which could be regarded as another thing and be made [111]the subject ofpredicates. It involves only the assertion that there are properties which belongto each separate thing, not that there are properties belonging to the whole ofthings collectively. The philosophy which I wish to advocate may be calledlogical atomism or absolute pluralism, because, while maintaining that there aremany things, it denies that there is a whole composed of those things. We shallsee, therefore, that philosophical propositions, instead of being concerned withthe whole of things collectively, are concerned with all things distributively;and not only must they be concerned with all things, but they must beconcerned with such properties of all things as do not depend upon theaccidental nature of the things that there happen to be, but are true of anypossible world, independently of such facts as can only be discovered by oursenses. This brings us to a second characteristic of philosophical propositions,namely, that they must be a priori. A philosophical proposition must be such ascan be neither proved nor disproved by empirical evidence. Too often we findin philosophical books arguments based upon the course of history, or theconvolutions of the brain, or the eyes of shell-fish. Special and accidental factsof this kind are irrelevant to philosophy, which must make only such assertionsas would be equally true however the actual world were constituted. We may sum up these two characteristics of philosophical propositions bysaying that philosophy is the science of the possible. But this statementunexplained is liable to be misleading, since it may be thought that the possibleis something other than the general, whereas in fact the two areindistinguishable. Philosophy, if what has been said is correct, becomes indistinguishable fromlogic as that word has now come [112]to be used. The study of logic consists,broadly speaking, of two not very sharply distinguished portions. On the onehand it is concerned with those general statements which can be madeconcerning everything without mentioning any one thing or predicate orrelation, such for example as "if x is a member of the class α and every memberof α is a member of β, then x is a member of the class β, whatever x, α, and βmay be." On the other hand, it is concerned with the analysis and enumerationof logical forms, i.e. with the kinds of propositions that may occur, with thevarious types of facts, and with the classification of the constituents of facts. Inthis way logic provides an inventory of possibilities, a repertory of abstractlytenable hypotheses. It might be thought that such a study would be too vague and too general tobe of any very great importance, and that, if its problems became at any pointsufficiently definite, they would be merged in the problems of some specialscience. It appears, however, that this is not the case. In some problems, forexample, the analysis of space and time, the nature of perception, or the theoryof judgment, the discovery of the logical form of the facts involved is thehardest part of the work and the part whose performance has been most lackinghitherto. It is chiefly for want of the right logical hypothesis that such problemshave hitherto been treated in such an unsatisfactory manner, and have givenrise to those contradictions or antinomies in which the enemies of reasonamong philosophers have at all times delighted. By concentrating attention upon the investigation of logical forms, itbecomes possible at last for philosophy to deal with its problems piecemeal,and to obtain, as the sciences do, such partial and probably not wholly correctresults as subsequent investigation can utilise [113]even while it supplements andimproves them. Most philosophies hitherto have been constructed all in oneblock, in such a way that, if they were not wholly correct, they were whollyincorrect, and could not be used as a basis for further investigations. It ischiefly owing to this fact that philosophy, unlike science, has hitherto beenunprogressive, because each original philosopher has had to begin the workagain from the beginning, without being able to accept anything definite fromthe work of his predecessors. A scientific philosophy such as I wish torecommend will be piecemeal and tentative like other sciences; above all, itwill be able to invent hypotheses which, even if they are not wholly true, willyet remain fruitful after the necessary corrections have been made. Thispossibility of successive approximations to the truth is, more than anythingelse, the source of the triumphs of science, and to transfer this possibility tophilosophy is to ensure a progress in method whose importance it would bealmost impossible to exaggerate. The essence of philosophy as thus conceived is analysis, not synthesis. Tobuild up systems of the world, like Heine's German professor who knit togetherfragments of life and made an intelligible system out of them, is not, I believe,any more feasible than the discovery of the philosopher's stone. What isfeasible is the understanding of general forms, and the division of traditionalproblems into a number of separate and less baffling questions. "Divide andconquer" is the maxim of success here as elsewhere. Let us illustrate these somewhat general maxims by examining theirapplication to the philosophy of space, for it is only in application that themeaning or importance of a method can be understood. Suppose we areconfronted with the problem of space as presented in [114]Kant's TranscendentalÆsthetic, and suppose we wish to discover what are the elements of theproblem and what hope there is of obtaining a solution of them. It will soonappear that three entirely distinct problems, belonging to different studies, andrequiring different methods for their solution, have been confusedly combinedin the supposed single problem with which Kant is concerned. There is aproblem of logic, a problem of physics, and a problem of theory of knowledge.Of these three, the problem of logic can be solved exactly and perfectly; theproblem of physics can probably be solved with as great a degree of certaintyand as great an approach to exactness as can be hoped in an empirical region;the problem of theory of knowledge, however, remains very obscure and verydifficult to deal with. Let us see how these three problems arise. (1) The logical problem has arisen through the suggestions of non-Euclideangeometry. Given a body of geometrical propositions, it is not difficult to find aminimum statement of the axioms from which this body of propositions can bededuced. It is also not difficult, by dropping or altering some of these axioms,to obtain a more general or a different geometry, having, from the point of viewof pure mathematics, the same logical coherence and the same title to respect asthe more familiar Euclidean geometry. The Euclidean geometry itself is trueperhaps of actual space (though this is doubtful), but certainly of an infinitenumber of purely arithmetical systems, each of which, from the point of viewof abstract logic, has an equal and indefeasible right to be called a Euclideanspace. Thus space as an object of logical or mathematical study loses itsuniqueness; not only are there many kinds of spaces, but there are an infinity ofexamples of each kind, [115]though it is difficult to find any kind of which thespace of physics may be an example, and it is impossible to find any kind ofwhich the space of physics is certainly an example. As an illustration of onepossible logical system of geometry we may consider all relations of threeterms which are analogous in certain formal respects to the relation "between"as it appears to be in actual space. A space is then defined by means of onesuch three-term relation. The points of the space are all the terms which havethis relation to something or other, and their order in the space in question isdetermined by this relation. The points of one space are necessarily also pointsof other spaces, since there are necessarily other three-term relations havingthose same points for their field. The space in fact is not determined by theclass of its points, but by the ordering three-term relation. When enoughabstract logical properties of such relations have been enumerated to determinethe resulting kind of geometry, say, for example, Euclidean geometry, itbecomes unnecessary for the pure geometer in his abstract capacity todistinguish between the various relations which have all these properties. Heconsiders the whole class of such relations, not any single one among them.Thus in studying a given kind of geometry the pure mathematician is studying acertain class of relations defined by means of certain abstract logical propertieswhich take the place of what used to be called axioms. The nature ofgeometrical reasoning therefore is purely deductive and purely logical; if anyspecial epistemological peculiarities are to be found in geometry, it must not bein the reasoning, but in our knowledge concerning the axioms in some givenspace. (2) The physical problem of space is both more interesting and more difficultthan the logical problem. [116]The physical problem may be stated as follows: tofind in the physical world, or to construct from physical materials, a space ofone of the kinds enumerated by the logical treatment of geometry. Thisproblem derives its difficulty from the attempt to accommodate to theroughness and vagueness of the real world some system possessing the logicalclearness and exactitude of pure mathematics. That this can be done with acertain degree of approximation is fairly evident If I see three people A, B,and C sitting in a row, I become aware of the fact which may be expressed bysaying that B is between A and C rather than that A is between B and C, or C isbetween A and B. This relation of "between" which is thus perceived to holdhas some of the abstract logical properties of those three-term relations which,we saw, give rise to a geometry, but its properties fail to be exact, and are not,as empirically given, amenable to the kind of treatment at which geometryaims. In abstract geometry we deal with points, straight lines, and planes; butthe three people A,B, and C whom I see sitting in a row are not exactly points,nor is the row exactly a straight line. Nevertheless physics, which formallyassumes a space containing points, straight lines, and planes, is foundempirically to give results applicable to the sensible world. It must therefore bepossible to find an interpretation of the points, straight lines, and planes ofphysics in terms of physical data, or at any rate in terms of data together withsuch hypothetical additions as seem least open to question. Since all data sufferfrom a lack of mathematical precision through being of a certain size andsomewhat vague in outline, it is plain that if such a notion as that of a point isto find any application to empirical material, the point must be neither a datumnor a hypothetical addition to [117]data, but a construction by means of data withtheir hypothetical additions. It is obvious that any hypothetical filling out ofdata is less dubious and unsatisfactory when the additions are closely analogousto data than when they are of a radically different sort. To assume, for example,that objects which we see continue, after we have turned away our eyes, to bemore or less analogous to what they were while we were looking, is a lessviolent assumption than to assume that such objects are composed of an infinitenumber of mathematical points. Hence in the physical study of the geometry ofphysical space, points must not be assumed ab initio as they are in the logicaltreatment of geometry, but must be constructed as systems composed of dataand hypothetical analogues of data. We are thus led naturally to define aphysical point as a certain class of those objects which are the ultimateconstituents of the physical world. It will be the class of all those objects which,as one would naturally say, contain the point. To secure a definition giving thisresult, without previously assuming that physical objects are composed ofpoints, is an agreeable problem in mathematical logic. The solution of thisproblem and the perception of its importance are due to my friend Dr.Whitehead. The oddity of regarding a point as a class of physical entities wearsoff with familiarity, and ought in any case not to be felt by those who maintain,as practically every one does, that points are mathematical fictions. The word"fiction" is used glibly in such connexions by many men who seem not to feelthe necessity of explaining how it can come about that a fiction can be so usefulin the study of the actual world as the points of mathematical physics have beenfound to be. By our definition, which regards a point as a class of physicalobjects, it is explained both how [118]the use of points can lead to importantphysical results, and how we can nevertheless avoid the assumption that pointsare themselves entities in the physical world. Many of the mathematically convenient properties of abstract logical spacescannot be either known to belong or known not to belong to the space ofphysics. Such are all the properties connected with continuity. For to know thatactual space has these properties would require an infinite exactness of sense-perception. If actual space is continuous, there are nevertheless many possiblenon-continuous spaces which will be empirically indistinguishable from it; and,conversely, actual space may be non-continuous and yet empiricallyindistinguishable from a possible continuous space. Continuity, therefore,though obtainable in the a priori region of arithmetic, is not with certaintyobtainable in the space or time of the physical world: whether these arecontinuous or not would seem to be a question not only unanswered but forever unanswerable. From the point of view of philosophy, however, thediscovery that a question is unanswerable is as complete an answer as any thatcould possibly be obtained. And from the point of view of physics, where noempirical means of distinction can be found, there can be no empiricalobjection to the mathematically simplest assumption, which is that ofcontinuity. The subject of the physical theory of space is a very large one, hitherto littleexplored. It is associated with a similar theory of time, and both have beenforced upon the attention of philosophically minded physicists by thediscussions which have raged concerning the theory of relativity. (3) The problem with which Kant is concerned in the TranscendentalÆsthetic is primarily the epistemological [119]problem: "How do we come tohave knowledge of geometry a priori?" By the distinction between the logicaland physical problems of geometry, the bearing and scope of this question aregreatly altered. Our knowledge of pure geometry is a priori but is whollylogical. Our knowledge of physical geometry is synthetic, but is not a priori.Our knowledge of pure geometry is hypothetical, and does not enable us toassert, for example, that the axiom of parallels is true in the physical world. Ourknowledge of physical geometry, while it does enable us to assert that thisaxiom is approximately verified, does not, owing to the inevitable inexactitudeof observation, enable us to assert that it is verified exactly. Thus, with theseparation which we have made between pure geometry and the geometry ofphysics, the Kantian problem collapses. To the question, "How is synthetic aprioriknowledge possible?" we can now reply, at any rate so far as geometry isconcerned, "It is not possible," if "synthetic" means "not deducible from logicalone." Our knowledge of geometry, like the rest of our knowledge, is derivedpartly from logic, partly from sense, and the peculiar position which in Kant'sday geometry appeared to occupy is seen now to be a delusion. There are stillsome philosophers, it is true, who maintain that our knowledge that the axiomof parallels, for example, is true of actual space, is not to be accounted forempirically, but is as Kant maintained derived from an a priori intuition. Thisposition is not logically refutable, but I think it loses all plausibility as soon aswe realise how complicated and derivative is the notion of physical space. Aswe have seen, the application of geometry to the physical world in no waydemands that there should really be points and straight lines among physicalentities. The principle of economy, [120]therefore, demands that we shouldabstain from assuming the existence of points and straight lines. As soon,however, as we accept the view that points and straight lines are complicatedconstructions by means of classes of physical entities, the hypothesis that wehave an a priori intuition enabling us to know what happens to straight lineswhen they are produced indefinitely becomes extremely strained and harsh; nordo I think that such an hypothesis would ever have arisen in the mind of aphilosopher who had grasped the nature of physical space. Kant, under theinfluence of Newton, adopted, though with some vacillation, the hypothesis ofabsolute space, and this hypothesis, though logically unobjectionable, isremoved by Occam's razor, since absolute space is an unnecessary entity in theexplanation of the physical world. Although, therefore, we cannot refute theKantian theory of an a priori intuition, we can remove its grounds one by onethrough an analysis of the problem. Thus, here as in many other philosophicalquestions, the analytic method, while not capable of arriving at a demonstrativeresult, is nevertheless capable of showing that all the positive grounds in favourof a certain theory are fallacious and that a less unnatural theory is capable ofaccounting for the facts. Another question by which the capacity of the analytic method can be shownis the question of realism. Both those who advocate and those who combatrealism seem to me to be far from clear as to the nature of the problem whichthey are discussing. If we ask: "Are our objects of perception real and arethey independent of the percipient?" it must be supposed that we attach somemeaning to the words "real" and "independent," and yet, if either side in thecontroversy of realism is asked to define these two words, their answer ispretty [121]sure to embody confusions such as logical analysis will reveal. Let us begin with the word "real." There certainly are objects of perception,and therefore, if the question whether these objects are real is to be a substantialquestion, there must be in the world two sorts of objects, namely, the real andthe unreal, and yet the unreal is supposed to be essentially what there is not.The question what properties must belong to an object in order to make it realis one to which an adequate answer is seldom if ever forthcoming. There is ofcourse the Hegelian answer, that the real is the self-consistent and that nothingis self-consistent except the Whole; but this answer, true or false, is not relevantin our present discussion, which moves on a lower plane and is concerned withthe status of objects of perception among other objects of equalfragmentariness. Objects of perception are contrasted, in the discussionsconcerning realism, rather with psychical states on the one hand and matter onthe other hand than with the all-inclusive whole of things. The question wehave therefore to consider is the question as to what can be meant by assigning"reality" to some but not all of the entities that make up the world. Twoelements, I think, make up what is felt rather than thought when the word"reality" is used in this sense. A thing is real if it persists at times when it is notperceived; or again, a thing is real when it is correlated with other things in away which experience has led us to expect. It will be seen that reality in eitherof these senses is by no means necessary to a thing, and that in fact there mightbe a whole world in which nothing was real in either of these senses. It mightturn out that the objects of perception failed of reality in one or both of theserespects, without its being in any way deducible that they are [122]not parts ofthe external world with which physics deals. Similar remarks will apply to theword "independent." Most of the associations of this word are bound up withideas as to causation which it is not now possible to maintain. A is independentof Bwhen B is not an indispensable part of the cause of A. But when it isrecognised that causation is nothing more than correlation, and that there arecorrelations of simultaneity as well as of succession, it becomes evident thatthere is no uniqueness in a series of casual antecedents of a given event, butthat, at any point where there is a correlation of simultaneity, we can pass fromone line of antecedents to another in order to obtain a new series of causalantecedents. It will be necessary to specify the causal law according to whichthe antecedents are to be considered. I received a letter the other day from acorrespondent who had been puzzled by various philosophical questions. Afterenumerating them he says: "These questions led me from Bonn to Strassburg,where I found Professor Simmel." Now, it would be absurd to deny that thesequestions caused his body to move from Bonn to Strassburg, and yet it must besupposed that a set of purely mechanical antecedents could also be found whichwould account for this transfer of matter from one place to another. Owing tothis plurality of causal series antecedent to a given event, the notion of thecausebecomes indefinite, and the question of independence becomes correspondinglyambiguous. Thus, instead of asking simply whether A is independent of B, weought to ask whether there is a series determined by such and such causal lawsleading from B to A. This point is important in connexion with the particularquestion of objects of perception. It may be that no objects quite like thosewhich we perceive ever exist unperceived; [123]in this case there will be a causallaw according to which objects of perception are not independent of beingperceived. But even if this be the case, it may nevertheless also happen thatthere are purely physical causal laws determining the occurrence of objectswhich are perceived by means of other objects which perhaps are notperceived. In that case, in regard to such causal laws objects of perception willbe independent of being perceived. Thus the question whether objects ofperception are independent of being perceived is, as it stands, indeterminate,and the answer will be yes or no according to the method adopted of making itdeterminate. I believe that this confusion has borne a very large part inprolonging the controversies on this subject, which might well have seemedcapable of remaining for ever undecided. The view which I should wish toadvocate is that objects of perception do not persist unchanged at times whenthey are not perceived, although probably objects more or less resembling themdo exist at such times; that objects of perception are part, and the onlyempirically knowable part, of the actual subject-matter of physics, and arethemselves properly to be called physical; that purely physical laws existdetermining the character and duration of objects of perception without anyreference to the fact that they are perceived; and that in the establishment ofsuch laws the propositions of physics do not presuppose any propositions ofpsychology or even the existence of mind. I do not know whether realistswould recognise such a view as realism. All that I should claim for it is, that itavoids difficulties which seem to me to beset both realism and idealism ashitherto advocated, and that it avoids the appeal which they have made to ideaswhich logical analysis shows to be ambiguous. A further defence andelaboration of [124]the positions which I advocate, but for which time is lackingnow, will be found indicated in my book on Our Knowledge of the ExternalWorld.[22] The adoption of scientific method in philosophy, if I am not mistaken,compels us to abandon the hope of solving many of the more ambitious andhumanly interesting problems of traditional philosophy. Some of these itrelegates, though with little expectation of a successful solution, to specialsciences, others it shows to be such as our capacities are essentially incapableof solving. But there remain a large number of the recognised problems ofphilosophy in regard to which the method advocated gives all those advantagesof division into distinct questions, of tentative, partial, and progressive advance,and of appeal to principles with which, independently of temperament, allcompetent students must agree. The failure of philosophy hitherto has been duein the main to haste and ambition: patience and modesty, here as in othersciences, will open the road to solid and durable progress.

[19]Bosanquet, Logic, ii, p. 211.

THE ULTIMATE CONSTITUENTS OF MATTER[23]

I wish to discuss in this article no less a question than the ancient

metaphysical query, "What is matter?" The question, "What is matter?" in sofar as it concerns philosophy, is, I think, already capable of an answer which inprinciple will be as complete as an answer can hope to be; that is to say, we canseparate the problem into an essentially soluble and an essentially insolubleportion, and we can now see how to solve the essentially soluble portion, atleast as regards its main outlines. It is these outlines which I wish to suggest inthe present article. My main position, which is realistic, is, I hope and believe,not remote from that of Professor Alexander, by whose writings on this subjectI have profited greatly.[24] It is also in close accord with that of Dr. Nunn.[25] Common sense is accustomed to the division of the world into mind andmatter. It is supposed by all who have never studied philosophy that thedistinction between mind and matter is perfectly clear and easy, that the two donot at any point overlap, and that only a fool or a philosopher could be in doubtas to whether any given entity is mental or material. This simplefaith [126]survives in Descartes and in a somewhat modified form in Spinoza,but with Leibniz it begins to disappear, and from his day to our own almostevery philosopher of note has criticised and rejected the dualism of commonsense. It is my intention in this article to defend this dualism; but beforedefending it we must spend a few moments on the reasons which haveprompted its rejection. Our knowledge of the material world is obtained by means of the senses, ofsight and touch and so on. At first it is supposed that things are just as theyseem, but two opposite sophistications soon destroy this naïve belief. On theone hand the physicists cut up matter into molecules, atoms, corpuscles, and asmany more such subdivisions as their future needs may make them postulate,and the units at which they arrive are uncommonly different from the visible,tangible objects of daily life. A unit of matter tends more and more to besomething like an electromagnetic field filling all space, though having itsgreatest intensity in a small region. Matter consisting of such elements is asremote from daily life as any metaphysical theory. It differs from the theoriesof metaphysicians only in the fact that its practical efficacy proves that itcontains some measure of truth and induces business men to invest money onthe strength of it; but, in spite of its connection with the money market, itremains a metaphysical theory none the less. The second kind of sophistication to which the world of common sense hasbeen subjected is derived from the psychologists and physiologists. Thephysiologists point out that what we see depends upon the eye, that what wehear depends upon the ear, and that all our senses are liable to be affected byanything which affects the brain, like alcohol or hasheesh. Psychologists pointout how much of what we think we see is supplied by association [127]orunconscious inference, how much is mental interpretation, and how doubtful isthe residuum which can be regarded as crude datum. From these facts it isargued by the psychologists that the notion of a datum passively received bythe mind is a delusion, and it is argued by the physiologists that even if a puredatum of sense could be obtained by the analysis of experience, still this datumcould not belong, as common sense supposes, to the outer world, since itswhole nature is conditioned by our nerves and sense organs, changing as theychange in ways which it is thought impossible to connect with any change inthe matter supposed to be perceived. This physiologist's argument is exposed tothe rejoinder, more specious than solid, that our knowledge of the existence ofthe sense organs and nerves is obtained by that very process which thephysiologist has been engaged in discrediting, since the existence of the nervesand sense organs is only known through the evidence of the senses themselves.This argument may prove that some reinterpretation of the results ofphysiology is necessary before they can acquire metaphysical validity. But itdoes not upset the physiological argument in so far as this constitutes merelya reductio ad absurdum of naïve realism. These various lines of argument prove, I think, that some part of the beliefsof common sense must be abandoned. They prove that, if we take these beliefsas a whole, we are forced into conclusions which are in part self-contradictory;but such arguments cannot of themselves decide what portion of our common-sense beliefs is in need of correction. Common sense believes that what we seeis physical, outside the mind, and continuing to exist if we shut our eyes or turnthem in another direction. I believe that common sense is right in [128]regardingwhat we see as physical and (in one of several possible senses) outside themind, but is probably wrong in supposing that it continues to exist when we areno longer looking at it. It seems to me that the whole discussion of matter hasbeen obscured by two errors which support each other. The first of these is theerror that what we see, or perceive through any of our other senses, issubjective: the second is the belief that what is physical must be persistent.Whatever physics may regard as the ultimate constituents of matter, it alwayssupposes these constituents to be indestructible. Since the immediate data ofsense are not indestructible but in a state of perpetual flux, it is argued thatthese data themselves cannot be among the ultimate constituents of matter. Ibelieve this to be a sheer mistake. The persistent particles of mathematicalphysics I regard as logical constructions, symbolic fictions enabling us toexpress compendiously very complicated assemblages of facts; and, on theother hand, I believe that the actual data in sensation, the immediate objects ofsight or touch or hearing, are extra-mental, purely physical, and among theultimate constituents of matter. My meaning in regard to the impermanence of physical entities may perhapsbe made clearer by the use of Bergson's favourite illustration of thecinematograph. When I first read Bergson's statement that the mathematicianconceives the world after the analogy of a cinematograph, I had never seen acinematograph, and my first visit to one was determined by the desire to verifyBergson's statement, which I found to be completely true, at least so far as I amconcerned. When, in a picture palace, we see a man rolling down hill, orrunning away from the police, or falling into a river, or doing any of those otherthings to which men in such places are addicted, we know[129]that there is notreally only one man moving, but a succession of films, each with a differentmomentary man. The illusion of persistence arises only through the approach tocontinuity in the series of momentary men. Now what I wish to suggest is thatin this respect the cinema is a better metaphysician than common sense,physics, or philosophy. The real man too, I believe, however the police mayswear to his identity, is really a series of momentary men, each different onefrom the other, and bound together, not by a numerical identity, but bycontinuity and certain intrinsic causal laws. And what applies to men appliesequally to tables and chairs, the sun, moon and stars. Each of these is to beregarded, not as one single persistent entity, but as a series of entitiessucceeding each other in time, each lasting for a very brief period, thoughprobably not for a mere mathematical instant. In saying this I am only urgingthe same kind of division in time as we are accustomed to acknowledge in thecase of space. A body which fills a cubic foot will be admitted to consist ofmany smaller bodies, each occupying only a very tiny volume; similarly a thingwhich persists for an hour is to be regarded as composed of many things of lessduration. A true theory of matter requires a division of things into time-corpuscles as well as into space-corpuscles. The world may be conceived as consisting of a multitude of entities arrangedin a certain pattern. The entities which are arranged I shall call "particulars."The arrangement or pattern results from relations among particulars. Classes orseries of particulars, collected together on account of some property whichmakes it convenient to be able to speak of them as wholes, are what I calllogical constructions or symbolic fictions. The particulars are to be conceived,not on the analogy of bricks [130]in a building, but rather on the analogy of notesin a symphony. The ultimate constituents of a symphony (apart from relations)are the notes, each of which lasts only for a very short time. We may collecttogether all the notes played by one instrument: these may be regarded as theanalogues of the successive particulars which common sense would regard assuccessive states of one "thing." But the "thing" ought to be regarded as nomore "real" or "substantial" than, for example, the rôle of the trombone. Assoon as "things" are conceived in this manner it will be found that thedifficulties in the way of regarding immediate objects of sense as physical havelargely disappeared. When people ask, "Is the object of sense mental or physical?" they seldomhave any clear idea either what is meant by "mental" or "physical," or whatcriteria are to be applied for deciding whether a given entity belongs to oneclass or the other. I do not know how to give a sharp definition of the word"mental," but something may be done by enumerating occurrences which areindubitably mental: believing, doubting, wishing, willing, being pleased orpained, are certainly mental occurrences; so are what we may call experiences,seeing, hearing, smelling, perceiving generally. But it does not follow from thisthat what is seen, what is heard, what is smelt, what is perceived, must bemental. When I see a flash of lightning, my seeing of it is mental, but what Isee, although it is not quite the same as what anybody else sees at the samemoment, and although it seems very unlike what the physicist would describeas a flash of lightning, is not mental. I maintain, in fact, that if the physicistcould describe truly and fully all that occurs in the physical world when there isa flash of lightning, it would contain as a constituent what I see, and alsowhat [131]is seen by anybody else who would commonly be said to see the sameflash. What I mean may perhaps be made plainer by saying that if my bodycould remain in exactly the same state in which it is, although my mind hadceased to exist, precisely that object which I now see when I see the flashwould exist, although of course I should not see it, since my seeing is mental.The principal reasons which have led people to reject this view have, I think,been two: first, that they did not adequately distinguish between my seeing andwhat I see; secondly, that the causal dependence of what I see upon my bodyhas made people suppose that what I see cannot be "outside" me. The first ofthese reasons need not detain us, since the confusion only needs to be pointedout in order to be obviated; but the second requires some discussion, since itcan only be answered by removing current misconceptions, on the one hand asto the nature of space, and on the other, as to the meaning of causaldependence. When people ask whether colours, for example, or other secondary qualitiesare inside or outside the mind, they seem to suppose that their meaning must beclear, and that it ought to be possible to say yes or no without any furtherdiscussion of the terms involved. In fact, however, such terms as "inside" or"outside" are very ambiguous. What is meant by asking whether this or that is"in" the mind? The mind is not like a bag or a pie; it does not occupy a certainregion in space, or, if (in a sense) it does, what is in that region is presumablypart of the brain, which would not be said to be in the mind. When people saythat sensible qualities are in the mind, they do not mean "spatially contained in"in the sense in which the blackbirds were in the pie. We might regard the mindas an assemblage of particulars, namely, what[132]would be called "states ofmind," which would belong together in virtue of some specific commonquality. The common quality of all states of mind would be the qualitydesignated by the word "mental"; and besides this we should have to supposethat each separate person's states of mind have some common characteristicdistinguishing them from the states of mind of other people. Ignoring this latterpoint, let us ask ourselves whether the quality designated by the word "mental"does, as a matter of observation, actually belong to objects of sense, such ascolours or noises. I think any candid person must reply that, however difficult itmay be to know what we mean by "mental," it is not difficult to see that coloursand noises are not mental in the sense of having that intrinsic peculiarity whichbelongs to beliefs and wishes and volitions, but not to the physical world.Berkeley advances on this subject a plausible argument[26] which seems to meto rest upon an ambiguity in the word "pain." He argues that the realistsupposes the heat which he feels in approaching a fire to be something outsidehis mind, but that as he approaches nearer and nearer to the fire the sensation ofheat passes imperceptibly into pain, and that no one could regard pain assomething outside the mind. In reply to this argument, it should be observed inthe first place that the heat of which we are immediately aware is not in the firebut in our own body. It is only by inference that the fire is judged to be thecause of the heat which we feel in our body. In the second place (and this is themore important point), when we speak of pain we may mean one of two things:we may mean the object of the sensation or other experience which has thequality of being painful, [133]or we may mean the quality of painfulness itself.When a man says he has a pain in his great toe, what he means is that he has asensation associated with his great toe and having the quality of painfulness.The sensation itself, like every sensation, consists in experiencing a sensibleobject, and the experiencing has that quality of painfulness which only mentaloccurrences can have, but which may belong to thoughts or desires, as well asto sensations. But in common language we speak of the sensible objectexperienced in a painful sensation as a pain, and it is this way of speakingwhich causes the confusion upon which the plausibility of Berkeley's argumentdepends. It would be absurd to attribute the quality of painfulness to anythingnon-mental, and hence it comes to be thought that what we call a pain in the toemust be mental. In fact, however, it is not the sensible object in such a casewhich is painful, but the sensation, that is to say, the experience of the sensibleobject. As the heat which we experience from the fire grows greater, theexperience passes gradually from being pleasant to being painful, but neitherthe pleasure nor the pain is a quality of the object experienced as opposed to theexperience, and it is therefore a fallacy to argue that this object must be mentalon the ground that painfulness can only be attributed to what is mental. If, then, when we say that something is in the mind we mean that it has acertain recognisable intrinsic characteristic such as belongs to thoughts anddesires, it must be maintained on grounds of immediate inspection that objectsof sense are not in any mind. A different meaning of "in the mind" is, however, to be inferred from thearguments advanced by those who regard sensible objects as being in the mind.The arguments used are, in the main, such as would prove the [134]causaldependence of objects of sense upon the percipient. Now the notion of causaldependence is very obscure and difficult, much more so in fact than is generallyrealised by philosophers. I shall return to this point in a moment. For thepresent, however, accepting the notion of causal dependence without criticism,I wish to urge that the dependence in question is rather upon our bodies thanupon our minds. The visual appearance of an object is altered if we shut oneeye, or squint, or look previously at something dazzling; but all these are bodilyacts, and the alterations which they effect are to be explained by physiologyand optics, not by psychology.[27] They are in fact of exactly the same kind asthe alterations effected by spectacles or a microscope. They belong therefore tothe theory of the physical world, and can have no bearing upon the questionwhether what we see is causally dependent upon the mind. What they do tendto prove, and what I for my part have no wish to deny, is that what we see iscausally dependent upon our body and is not, as crude common sense wouldsuppose, something which would exist equally if our eyes and nerves and brainwere absent, any more than the visual appearance presented by an object seenthrough a microscope would remain if the microscope were removed. So longas it is supposed that the physical world is composed of stable and more or lesspermanent constituents, the fact that what we see is changed by changes in ourbody appears to afford reason for regarding what we see as not an ultimateconstituent of matter. But if it is recognised that the ultimate constituents ofmatter are as circumscribed in duration as in spatial extent, the whole of thisdifficulty vanishes. There remains, however, another difficulty, connected with space. When welook at the sun we wish to know [135]something about the sun itself, which isninety-three million miles away; but what we see is dependent upon our eyes,and it is difficult to suppose that our eyes can affect what happens at a distanceof ninety-three million miles. Physics tells us that certain electromagneticwaves start from the sun, and reach our eyes after about eight minutes. Theythere produce disturbances in the rods and cones, thence in the optic nerve,thence in the brain. At the end of this purely physical series, by some oddmiracle, comes the experience which we call "seeing the sun," and it is suchexperiences which form the whole and sole reason for our belief in the opticnerve, the rods and cones, the ninety-three million miles, the electromagneticwaves, and the sun itself. It is this curious oppositeness of direction betweenthe order of causation as affirmed by physics, and the order of evidence asrevealed by theory of knowledge, that causes the most serious perplexities inregard to the nature of physical reality. Anything that invalidates our seeing, asa source of knowledge concerning physical reality, invalidates also the wholeof physics and physiology. And yet, starting from a common-sense acceptanceof our seeing, physics has been led step by step to the construction of the causalchain in which our seeing is the last link, and the immediate object which wesee cannot be regarded as that initial cause which we believe to be ninety-threemillion miles away, and which we are inclined to regard as the "real" sun. I have stated this difficulty as forcibly as I can, because I believe that it canonly be answered by a radical analysis and reconstruction of all the conceptionsupon whose employment it depends. Space, time, matter and cause, are the chief of these conceptions. Let usbegin with the conception of cause. Causal dependence, as I observed a moment ago, is a [136]conception which itis very dangerous to accept at its face value. There exists a notion that in regardto any event there is something which may be called the cause of that event—some one definite occurrence, without which the event would have beenimpossible and with which it becomes necessary. An event is supposed to bedependent upon its cause in some way which in it is not dependent upon otherthings. Thus men will urge that the mind is dependent upon the brain, or, withequal plausibility, that the brain is dependent upon the mind. It seems notimprobable that if we had sufficient knowledge we could infer the state of aman's mind from the state of his brain, or the state of his brain from the state ofhis mind. So long as the usual conception of causal dependence is retained, thisstate of affairs can be used by the materialist to urge that the state of our braincauses our thoughts, and by the idealist to urge that our thoughts cause the stateof our brain. Either contention is equally valid or equally invalid. The factseems to be that there are many correlations of the sort which may be calledcausal, and that, for example, either a physical or a mental event can bepredicted, theoretically, either from a sufficient number of physical antecedentsor from a sufficient number of mental antecedents. To speak of the cause of anevent is therefore misleading. Any set of antecedents from which the event cantheoretically be inferred by means of correlations might be called a cause of theevent. But to speak of the cause is to imply a uniqueness which does not exist. The relevance of this to the experience which we call "seeing the sun" isobvious. The fact that there exists a chain of antecedents which makes ourseeing dependent upon the eyes and nerves and brain does not even tend toshow that there is not another chain of antecedents in which the eyes and nervesand brain as physical things are ignored. If we are to escape from the dilemmawhich [137]seemed to arise out of the physiological causation of what we seewhen we say we see the sun, we must find, at least in theory, a way of statingcausal laws for the physical world, in which the units are not material things,such as the eyes and nerves and brain, but momentary particulars of the samesort as our momentary visual object when we look at the sun. The sun itself andthe eyes and nerves and brain must be regarded as assemblages of momentaryparticulars. Instead of supposing, as we naturally do when we start from anuncritical acceptance of the apparent dicta of physics, that matter is what is"really real" in the physical world, and that the immediate objects of sense aremere phantasms, we must regard matter as a logical construction, of which theconstituents will be just such evanescent particulars as may, when an observerhappens to be present, become data of sense to that observer. What physicsregards as the sun of eight minutes ago will be a whole assemblage ofparticulars, existing at different times, spreading out from a centre with thevelocity of light, and containing among their number all those visual datawhich are seen by people who are now looking at the sun. Thus the sun of eightminutes ago is a class of particulars, and what I see when I now look at the sunis one member of this class. The various particulars constituting this class willbe correlated with each other by a certain continuity and certain intrinsic lawsof variation as we pass outwards from the centre, together with certainmodifications correlated extrinsically with other particulars which are notmembers of this class. It is these extrinsic modifications which represent thesort of facts that, in our former account, appeared as the influence of the eyesand nerves in modifying the appearance of the sun.[28] [138]The prima facie difficulties in the way of this view are chiefly derivedfrom an unduly conventional theory of space. It might seem at first sight as ifwe had packed the world much fuller than it could possibly hold. At everyplace between us and the sun, we said, there is to be a particular which is to bea member of the sun as it was a few minutes ago. There will also, of course,have to be a particular which is a member of any planet or fixed star that mayhappen to be visible from that place. At the place where I am, there will beparticulars which will be members severally of all the "things" I am now said tobe perceiving. Thus throughout the world, everywhere, there will be anenormous number of particulars coexisting in the same place. But thesetroubles result from contenting ourselves too readily with the merely three-dimensional space to which schoolmasters have accustomed us. The space ofthe real world is a space of six dimensions, and as soon as we realise this wesee that there is plenty of room for all the particulars for which we want to findpositions. In order to realise this we have only to return for a moment from thepolished space of physics to the rough and untidy space of our immediatesensible experience. The space of one man's sensible objects is a three-dimensional space. It does not appear probable that two men ever both perceiveat the same time any one sensible object; when they are said to see the samething or hear the same noise, there will always be some difference, howeverslight, between the actual shapes seen or the actual sounds heard. If this is so,and if, as is generally assumed, position in space is purely relative, it followsthat the space of one man's objects and the space of another man's objects haveno place in common, that they are in fact different spaces, and not merelydifferent parts of one space. I mean by this that such immediate spatial relationsas are perceived to hold [139]between the different parts of the sensible spaceperceived by one man, do not hold between parts of sensible spaces perceivedby different men. There are therefore a multitude of three-dimensional spacesin the world: there are all those perceived by observers, and presumably alsothose which are not perceived, merely because no observer is suitably situatedfor perceiving them. But although these spaces do not have to one another the same kind of spatialrelations as obtain between the parts of one of them, it is nevertheless possibleto arrange these spaces themselves in a three-dimensional order. This is doneby means of the correlated particulars which we regard as members (or aspects)of one physical thing. When a number of people are said to see the same object,those who would be said to be near to the object see a particular occupying alarger part of their field of vision than is occupied by the correspondingparticular seen by people who would be said to be farther from the thing. Bymeans of such considerations it is possible, in ways which need not now befurther specified, to arrange all the different spaces in a three-dimensionalseries. Since each of the spaces is itself three-dimensional, the whole world ofparticulars is thus arranged in a six-dimensional space, that is to say, six co-ordinates will be required to assign completely the position of any givenparticular, namely, three to assign its position in its own space and three moreto assign the position of its space among the other spaces. There are two ways of classifying particulars: we may take together all thosethat belong to a given "perspective," or all those that are, as common sensewould say, different "aspects" of the same "thing." For example, if I am (as issaid) seeing the sun, what I see belongs to two assemblages: (1) the assemblageof all my present objects of sense, which is what I call a "perspective"; [140](2)the assemblage of all the different particulars which would be called aspects ofthe sun of eight minutes ago—this assemblage is what I define as being the sunof eight minutes ago. Thus "perspectives" and "things" are merely two differentways of classifying particulars. It is to be observed that there is no apriori necessity for particulars to be susceptible of this double classification.There may be what might be called "wild" particulars, not having the usualrelations by which the classification is effected; perhaps dreams andhallucinations are composed of particulars which are "wild" in this sense. The exact definition of what is meant by a perspective is not quite easy. Solong as we confine ourselves to visible objects or to objects of touch we mightdefine the perspective of a given particular as "all particulars which have asimple (direct) spatial relation to the given particular." Between two patches ofcolour which I see now, there is a direct spatial relation which I equally see.But between patches of colour seen by different men there is only an indirectconstructed spatial relation by means of the placing of "things" in physicalspace (which is the same as the space composed of perspectives). Thoseparticulars which have direct spatial relations to a given particular will belongto the same perspective. But if, for example, the sounds which I hear are tobelong to the same perspective with the patches of colour which I see, theremust be particulars which have no direct spatial relation and yet belong to thesame perspective. We cannot define a perspective as all the data of onepercipient at one time, because we wish to allow the possibility of perspectiveswhich are not perceived by any one. There will be need, therefore, in defining aperspective, of some principle derived neither from psychology nor from space. Such a principle may be obtained from the [141]consideration of time. The oneall-embracing time, like the one all-embracing space, is a construction; there isno directtime-relation between particulars belonging to my perspective andparticulars belonging to another man's. On the other hand, any two particularsof which I am aware are either simultaneous or successive, and theirsimultaneity or successiveness is sometimes itself a datum to me. We maytherefore define the perspective to which a given particular belongs as "allparticulars simultaneous with the given particular," where "simultaneous" is tobe understood as a direct simple relation, not the derivative constructed relationof physics. It may be observed that the introduction of "local time" suggestedby the principle of relativity has effected, for purely scientific reasons, muchthe same multiplication of times as we have just been advocating. The sum-total of all the particulars that are (directly) either simultaneouswith or before or after a given particular may be defined as the "biography" towhich that particular belongs. It will be observed that, just as a perspectiveneed not be actually perceived by any one, so a biography need not be actuallylived by any one. Those biographies that are lived by no one are called"official." The definition of a "thing" is effected by means of continuity and ofcorrelations which have a certain differential independence of other "things."That is to say, given a particular in one perspective, there will usually in aneighbouring perspective be a very similar particular, differing from the givenparticular, to the first order of small quantities, according to a law involvingonly the difference of position of the two perspectives in perspective space, andnot any of the other "things" in the universe. It is this continuity and differentialindependence in the law of change as we pass from one [142]perspective toanother that defines the class of particulars which is to be called "one thing." Broadly speaking, we may say that the physicist finds it convenient toclassify particulars into "things," while the psychologist finds it convenient toclassify them into "perspectives" and "biographies," since oneperspective may constitute the momentary data of one percipient, and onebiography may constitute the whole of the data of one percipient throughout hislife. We may now sum up our discussion. Our object has been to discover as faras possible the nature of the ultimate constituents of the physical world. When Ispeak of the "physical world," I mean, to begin with, the world dealt with byphysics. It is obvious that physics is an empirical science, giving us a certainamount of knowledge and based upon evidence obtained through the senses.But partly through the development of physics itself, partly through argumentsderived from physiology, psychology or metaphysics, it has come to be thoughtthat the immediate data of sense could not themselves form part of the ultimateconstituents of the physical world, but were in some sense "mental," "in themind," or "subjective." The grounds for this view, in so far as they depend uponphysics, can only be adequately dealt with by rather elaborate constructionsdepending upon symbolic logic, showing that out of such materials as areprovided by the senses it is possible to construct classes and series having theproperties which physics assigns to matter. Since this argument is difficult andtechnical, I have not embarked upon it in this article. But in so far as the viewthat sense-data are "mental" rests upon physiology, psychology, ormetaphysics, I have tried to show that it rests upon confusions and prejudices—prejudices in favour of permanence in the ultimate constituents of matter,and [143]confusions derived from unduly simple notions as to space, from thecausal correlation of sense-data with sense-organs, and from failure todistinguish between sense-data and sensations. If what we have said on thesesubjects is valid, the existence of sense-data is logically independent of theexistence of mind, and is causally dependent upon the body of the percipientrather than upon his mind. The causal dependence upon the body of thepercipient, we found, is a more complicated matter than it appears to be, and,like all causal dependence, is apt to give rise to erroneous beliefs throughmisconceptions as to the nature of causal correlation. If we have been right inour contentions, sense-data are merely those among the ultimate constituents ofthe physical world, of which we happen to be immediately aware; theythemselves are purely physical, and all that is mental in connection with them isour awareness of them, which is irrelevant to their nature and to their place inphysics. Unduly simple notions as to space have been a great stumbling-block torealists. When two men look at the same table, it is supposed that what the onesees and what the other sees are in the same place. Since the shape and colourare not quite the same for the two men, this raises a difficulty, hastily solved, orrather covered up, by declaring what each sees to be purely "subjective"—though it would puzzle those who use this glib word to say what they mean byit. The truth seems to be that space—and time also—is much more complicatedthan it would appear to be from the finished structure of physics, and that theone all-embracing three-dimensional space is a logical construction, obtainedby means of correlations from a crude space of six dimensions. The particularsoccupying this six-dimensional space, classified in one way, form "things,"from which with certain further manipulations we can obtain what physicscan [144]regard as matter; classified in another way, they form "perspectives"and "biographies," which may, if a suitable percipient happens to exist, formrespectively the sense-data of a momentary or of a total experience. It is onlywhen physical "things" have been dissected into series of classes of particulars,as we have done, that the conflict between the point of view of physics and thepoint of view of psychology can be overcome. This conflict, if what has beensaid is not mistaken, flows from different methods of classification, andvanishes as soon as its source is discovered. In favour of the theory which I have briefly outlined, I do not claim that itis certainly true. Apart from the likelihood of mistakes, much of it is avowedlyhypothetical. What I do claim for the theory is that it may be true, and that thisis more than can be said for any other theory except the closely analogoustheory of Leibniz. The difficulties besetting realism, the confusions obstructingany philosophical account of physics, the dilemma resulting from discreditingsense-data, which yet remain the sole source of our knowledge of the outerworld—all these are avoided by the theory which I advocate. This does notprove the theory to be true, since probably many other theories might beinvented which would have the same merits. But it does prove that the theoryhas a better chance of being true than any of its present competitors, and itsuggests that what can be known with certainty is likely to be discoverable bytaking our theory as a starting-point, and gradually freeing it from all suchassumptions as seem irrelevant, unnecessary, or unfounded. On these grounds, Irecommend it to attention as a hypothesis and a basis for further work, thoughnot as itself a finished or adequate solution of the problem with which it deals. FOOTNOTES:

THE RELATION OF SENSE-DATA TO PHYSICS

I. THE PROBLEM STATED

Physics is said to be an empirical science, based upon observation and

experiment. It is supposed to be verifiable, i.e. capable of calculating beforehand resultssubsequently confirmed by observation and experiment. What can we learn by observation and experiment? Nothing, so far as physics is concerned, except immediate data of sense:certain patches of colour, sounds, tastes, smells, etc., with certain spatio-temporal relations. The supposed contents of the physical world are prima facie very differentfrom these: molecules have no colour, atoms make no noise, electrons have notaste, and corpuscles do not even smell. If such objects are to be verified, it must be solely through their relation tosense-data: they must have some kind of correlation with sense-data, and mustbe verifiable through their correlation alone. But how is the correlation itself ascertained? A correlation can only beascertained empirically by the correlated objects beingconstantly found together. But in our case, only one term of the correlation,namely, the sensible term, is ever found: the other term seems [146]essentiallyincapable of being found. Therefore, it would seem, the correlation with objectsof sense, by which physics was to be verified, is itself utterly and for everunverifiable. There are two ways of avoiding this result. (1) We may say that we know some principle a priori, without the need ofempirical verification, e.g. that our sense-data have causes other thanthemselves, and that something can be known about these causes by inferencefrom their effects. This way has been often adopted by philosophers. It may benecessary to adopt this way to some extent, but in so far as it is adopted physicsceases to be empirical or based upon experiment and observation alone.Therefore this way is to be avoided as much as possible. (2) We may succeed in actually defining the objects of physics as functionsof sense-data. Just in so far as physics leads to expectations, this must bepossible, since we can only expect what can be experienced. And in so far asthe physical state of affairs is inferred from sense-data, it must be capable ofexpression as a function of sense-data. The problem of accomplishing thisexpression leads to much interesting logico-mathematical work. In physics as commonly set forth, sense-data appear as functions of physicalobjects: when such-and-such waves impinge upon the eye, we see such-and-such colours, and so on. But the waves are in fact inferred from the colours, notvice versa. Physics cannot be regarded as validly based upon empirical datauntil the waves have been expressed as functions of the colours and othersense-data. Thus if physics is to be verifiable we are faced with the following problem:Physics exhibits sense-data as functions of physical objects, but verification isonly possible if physical objects can be exhibited as functions of [147]sense-data.We have therefore to solve the equations giving sense-data in terms of physicalobjects, so as to make them instead give physical objects in terms of sense-data. II. CHARACTERISTICS OF SENSE-DATA

When I speak of a "sense-datum," I do not mean the whole of what is given

in sense at one time. I mean rather such a part of the whole as might be singledout by attention: particular patches of colour, particular noises, and so on.There is some difficulty in deciding what is to be considered one sense-datum:often attention causes divisions to appear where, so far as can be discovered,there were no divisions before. An observed complex fact, such as that thispatch of red is to the left of that patch of blue, is also to be regarded as a datumfrom our present point of view: epistemologically, it does not differ greatlyfrom a simple sense-datum as regards its function in giving knowledge.Its logical structure is very different, however, from that of sense: sense givesacquaintance with particulars, and is thus a two-term relation in which theobject can be named but not asserted, and is inherently incapable of truth orfalsehood, whereas the observation of a complex fact, which may be suitablycalled perception, is not a two-term relation, but involves the propositionalform on the object-side, and gives knowledge of a truth, not mere acquaintancewith a particular. This logical difference, important as it is, is not very relevantto our present problem; and it will be convenient to regard data of perception asincluded among sense-data for the purposes of this paper. It is to be observedthat the particulars which are constituents of a datum of perception are alwayssense-data in the strict sense. [148]Concerning sense-data, we know that they are there while they are data,and this is the epistemological basis of all our knowledge of externalparticulars. (The meaning of the word "external" of course raises problemswhich will concern us later.) We do not know, except by means of more or lessprecarious inferences, whether the objects which are at one time sense-datacontinue to exist at times when they are not data. Sense-data at the times whenthey are data are all that we directly and primitively know of the externalworld; hence in epistemology the fact that they are data is all-important. Butthe fact that they are all that we directly know gives, of course, no presumptionthat they are all that there is. If we could construct an impersonal metaphysic,independent of the accidents of our knowledge and ignorance, the privilegedposition of the actual data would probably disappear, and they would probablyappear as a rather haphazard selection from a mass of objects more or less likethem. In saying this, I assume only that it is probable that there are particularswith which we are not acquainted. Thus the special importance of sense-data isin relation to epistemology, not to metaphysics. In this respect, physics is to bereckoned as metaphysics: it is impersonal, and nominally pays no specialattention to sense-data. It is only when we ask how physics can be known thatthe importance of sense-data re-emerges.

III. SENSIBILIA

I shall give the name sensibilia to those objects which have the samemetaphysical and physical status as sense-data, without necessarily being datato any mind. Thus the relation of a sensibile to a sense-datum is like that of aman to a husband: a man becomes a husband by [149]entering into the relation ofmarriage, and similarly asensibile becomes a sense-datum by entering into therelation of acquaintance. It is important to have both terms; for we wish todiscuss whether an object which is at one time a sense-datum can still exist at atime when it is not a sense-datum. We cannot ask "Can sense-data exist withoutbeing given?" for that is like asking "Can husbands exist without beingmarried?" We must ask "Can sensibilia exist without being given?" and also"Can a particular sensibile be at one time a sense-datum, and at another not?"Unless we have the word sensibile as well as the word "sense-datum," suchquestions are apt to entangle us in trivial logical puzzles. It will be seen that all sense-data are sensibilia. It is a metaphysical questionwhether all sensibilia are sense-data, and an epistemological question whetherthere exist means of inferring sensibilia which are not data from those that are. A few preliminary remarks, to be amplified as we proceed, will serve toelucidate the use which I propose to make of sensibilia. I regard sense-data as not mental, and as being, in fact, part of the actualsubject-matter of physics. There are arguments, shortly to be examined, fortheir subjectivity, but these arguments seem to me only toprove physiological subjectivity, i.e. causal dependence on the sense-organs,nerves, and brain. The appearance which a thing presents to us is causallydependent upon these, in exactly the same way as it is dependent uponintervening fog or smoke or coloured glass. Both dependences are contained inthe statement that the appearance which a piece of matter presents when viewedfrom a given place is a function not only of the piece of matter, but also of theintervening medium. (The terms used in [150]this statement—"matter," "viewfrom a given place," "appearance," "intervening medium"—will all be definedin the course of the present paper.) We have not the means of ascertaining howthings appear from places not surrounded by brain and nerves and sense-organs, because we cannot leave the body; but continuity makes it notunreasonable to suppose that they present some appearance at such places. Anysuch appearance would be included among sensibilia. If—per impossibile—there were a complete human body with no mind inside it, allthose sensibilia would exist, in relation to that body, which would be sense-dataif there were a mind in the body. What the mind adds to sensibilia, in fact,is merely awareness: everything else is physical or physiological.

IV. SENSE-DATA ARE PHYSICAL

Before discussing this question it will be well to define the sense in which theterms "mental" and "physical" are to be used. The word "physical," in allpreliminary discussions, is to be understood as meaning "what is dealt with byphysics." Physics, it is plain, tells us something about some of the constituentsof the actual world; what these constituents are may be doubtful, but it is theythat are to be called physical, whatever their nature may prove to be. The definition of the term "mental" is more difficult, and can only besatisfactorily given after many difficult controversies have been discussed anddecided. For present purposes therefore I must content myself with assuming adogmatic answer to these controversies. I shall call a particular "mental" whenit is aware of something, and I shall call a fact "mental" when it contains amental particular as a constituent. [151]It will be seen that the mental and the physical are not necessarilymutually exclusive, although I know of no reason to suppose that they overlap. The doubt as to the correctness of our definition of the "mental" is of littleimportance in our present discussion. For what I am concerned to maintain isthat sense-data are physical, and this being granted it is a matter of indifferencein our present inquiry whether or not they are also mental. Although I do nothold, with Mach and James and the "new realists," that the difference betweenthe mental and the physical is merely one of arrangement, yet what I have tosay in the present paper is compatible with their doctrine and might have beenreached from their standpoint. In discussions on sense-data, two questions are commonly confused, namely: (1) Do sensible objects persist when we are not sensible of them? in otherwords, do sensibilia which are data at a certain time sometimes continue toexist at times when they are not data? And (2) are sense-data mental orphysical? I propose to assert that sense-data are physical, while yet maintaining thatthey probably never persist unchanged after ceasing to be data. The view thatthey do not persist is often thought, quite erroneously in my opinion, to implythat they are mental; and this has, I believe, been a potent source of confusionin regard to our present problem. If there were, as some have held, a logicalimpossibility in sense-data persisting after ceasing to be data, that certainlywould tend to show that they were mental; but if, as I contend, their non-persistence is merely a probable inference from empirically ascertained causallaws, then it carries no such implication with it, and we are quite free to treatthem as part of the subject-matter of physics. [152]Logically a sense-datum is an object, a particular of which the subject isaware. It does not contain the subject as a part, as for example beliefs andvolitions do. The existence of the sense-datum is therefore not logicallydependent upon that of the subject; for the only way, so far as I know, in whichthe existence of A can belogically dependent upon the existence of B iswhen B is part of A. There is therefore no a priori reason why a particularwhich is a sense-datum should not persist after it has ceased to be a datum, norwhy other similar particulars should not exist without ever being data. Theview that sense-data are mental is derived, no doubt, in part from theirphysiological subjectivity, but in part also from a failure to distinguish betweensense-data and "sensations." By a sensation I mean the fact consisting in thesubject's awareness of the sense-datum. Thus a sensation is a complex of whichthe subject is a constituent and which therefore is mental. The sense-datum, onthe other hand, stands over against the subject as that external object of whichin sensation the subject is aware. It is true that the sense-datum is in many casesin the subject's body, but the subject's body is as distinct from the subject astables and chairs are, and is in fact merely a part of the material world. So soon,therefore, as sense-data are clearly distinguished from sensations, and as theirsubjectivity is recognised to be physiological not psychical, the chief obstaclesin the way of regarding them as physical are removed.

V. "SENSIBILIA" AND "THINGS"

But if "sensibilia" are to be recognised as the ultimate constituents of the

physical world, a long and difficult journey is to be performed before we canarrive either at[153]the "thing" of common sense or at the "matter" of physics.The supposed impossibility of combining the different sense-data which areregarded as appearances of the same "thing" to different people has made itseem as though these "sensibilia" must be regarded as mere subjectivephantasms. A given table will present to one man a rectangular appearance,while to another it appears to have two acute angles and two obtuse angles; toone man it appears brown, while to another, towards whom it reflects the light,it appears white and shiny. It is said, not wholly without plausibility, that thesedifferent shapes and different colours cannot co-exist simultaneously in thesame place, and cannot therefore both be constituents of the physical world.This argument I must confess appeared to me until recently to be irrefutable.The contrary opinion has, however, been ably maintained by Dr. T.P. Nunn inan article entitled: "Are Secondary Qualities Independent ofPerception?"[29] The supposed impossibility derives its apparent force from thephrase: "in the same place," and it is precisely in this phrase that its weaknesslies. The conception of space is too often treated in philosophy—even by thosewho on reflection would not defend such treatment—as though it were asgiven, simple, and unambiguous as Kant, in his psychological innocence,supposed. It is the unperceived ambiguity of the word "place" which, as weshall shortly see, has caused the difficulties to realists and given an undeservedadvantage to their opponents. Two "places" of different kinds are involved inevery sense-datum, namely the place at which it appears and theplace fromwhich it appears. These belong to different spaces, although, as weshall see, it is possible, with certain limitations, to establish a correlationbetween them. [154]What we call the different appearances of the same thing todifferent observers are each in a space private to the observer concerned. Noplace in the private world of one observer is identical with a place in the privateworld of another observer. There is therefore no question of combining thedifferent appearances in the one place; and the fact that they cannot all exist inone place affords accordingly no ground whatever for questioning theirphysical reality. The "thing" of common sense may in fact be identified withthe whole class of its appearances—where, however, we must include amongappearances not only those which are actual sense-data, but also those"sensibilia," if any, which, on grounds of continuity and resemblance, are to beregarded as belonging to the same system of appearances, although therehappen to be no observers to whom they are data. An example may make this clearer. Suppose there are a number of people ina room, all seeing, as they say, the same tables and chairs, walls and pictures.No two of these people have exactly the same sense-data, yet there is sufficientsimilarity among their data to enable them to group together certain of thesedata as appearances of one "thing" to the several spectators, and others asappearances of another "thing." Besides the appearances which a given thing inthe room presents to the actual spectators, there are, we may suppose, otherappearances which it would present to other possible spectators. If a man wereto sit down between two others, the appearance which the room would presentto him would be intermediate between the appearances which it presents to thetwo others: and although this appearance would not exist as it is without thesense organs, nerves and brain, of the newly arrived spectator, still it is notunnatural to suppose that, from the position [155]which he nowoccupies, some appearance of the room existed before his arrival. Thissupposition, however, need merely be noticed and not insisted upon. Since the "thing" cannot, without indefensible partiality, be identified withany single one of its appearances, it came to be thought of as something distinctfrom all of them and underlying them. But by the principle of Occam's razor, ifthe class of appearances will fulfil the purposes for the sake of which the thingwas invented by the prehistoric metaphysicians to whom common sense is due,economy demands that we should identify the thing with the class of itsappearances. It is not necessary todeny a substance or substratum underlyingthese appearances; it is merely expedient to abstain from asserting thisunnecessary entity. Our procedure here is precisely analogous to that which hasswept away from the philosophy of mathematics the useless menagerie ofmetaphysical monsters with which it used to be infested.

VI. CONSTRUCTIONS VERSUS INFERENCES

Before proceeding to analyse and explain the ambiguities of the word

"place," a few general remarks on method are desirable. The supreme maxim inscientific philosophising is this: Wherever possible, logical constructions are to be substituted for inferredentities. Some examples of the substitution of construction for inference in the realmof mathematical philosophy may serve to elucidate the uses of this maxim.Take first the case of irrationals. In old days, irrationals were inferred as thesupposed limits of series of rationals which had no rational limit; but theobjection to this procedure was[156]that it left the existence of irrationals merelyoptative, and for this reason the stricter methods of the present day no longertolerate such a definition. We now define an irrational number as a certain classof ratios, thus constructing it logically by means of ratios, instead of arriving atit by a doubtful inference from them. Take again the case of cardinal numbers.Two equally numerous collections appear to have something in common: thissomething is supposed to be their cardinal number. But so long as the cardinalnumber is inferred from the collections, not constructed in terms of them, itsexistence must remain in doubt, unless in virtue of a metaphysical postulate adhoc. By defining the cardinal number of a given collection as the class of allequally numerous collections, we avoid the necessity of this metaphysicalpostulate, and thereby remove a needless element of doubt from the philosophyof arithmetic. A similar method, as I have shown elsewhere, can be applied toclasses themselves, which need not be supposed to have any metaphysicalreality, but can be regarded as symbolically constructed fictions. The method by which the construction proceeds is closely analogous in theseand all similar cases. Given a set of propositions nominally dealing with thesupposed inferred entities, we observe the properties which are required of thesupposed entities in order to make these propositions true. By dint of a littlelogical ingenuity, we then construct some logical function of less hypotheticalentities which has the requisite properties. This constructed function wesubstitute for the supposed inferred entities, and thereby obtain a new and lessdoubtful interpretation of the body of propositions in question This method, sofruitful in the philosophy of mathematics, will be found equally applicable inthe philosophy of [157]physics, where, I do not doubt, it would have beenapplied long ago but for the fact that all who have studied this subject hithertohave been completely ignorant of mathematical logic. I myself cannot claimoriginality in the application of this method to physics, since I owe thesuggestion and the stimulus for its application entirely to my friend andcollaborator Dr. Whitehead, who is engaged in applying it to the moremathematical portions of the region intermediate between sense-data and thepoints, instants and particles of physics. A complete application of the method which substitutes constructions forinferences would exhibit matter wholly in terms of sense-data, and even, wemay add, of the sense-data of a single person, since the sense-data of otherscannot be known without some element of inference. This, however, mustremain for the present an ideal, to be approached as nearly as possible, but to bereached, if at all, only after a long preliminary labour of which as yet we canonly see the very beginning. The inferences which are unavoidable can,however, be subjected to certain guiding principles. In the first place theyshould always be made perfectly explicit, and should be formulated in the mostgeneral manner possible. In the second place the inferred entities should,whenever this can be done, be similar to those whose existence is given, ratherthan, like the Kantian Ding an sich, something wholly remote from the datawhich nominally support the inference. The inferred entities which I shall allowmyself are of two kinds: (a) the sense-data of other people, in favour of whichthere is the evidence of testimony, resting ultimately upon the analogicalargument in favour of minds other than my own; (b) the "sensibilia" whichwould appear from places where there happen to be no minds, and which Isuppose to be real although they are no one's [158]data. Of these two classes ofinferred entities, the first will probably be allowed to pass unchallenged. Itwould give me the greatest satisfaction to be able to dispense with it, and thusestablish physics upon a solipsistic basis; but those—and I fear they are themajority—in whom the human affections are stronger than the desire forlogical economy, will, no doubt, not share my desire to render solipsismscientifically satisfactory. The second class of inferred entities raises muchmore serious questions. It may be thought monstrous to maintain that a thingcan present any appearance at all in a place where no sense organs and nervousstructure exist through which it could appear. I do not myself feel themonstrosity; nevertheless I should regard these supposed appearances only inthe light of a hypothetical scaffolding, to be used while the edifice of physics isbeing raised, though possibly capable of being removed as soon as the edifice iscompleted. These "sensibilia" which are not data to anyone are therefore to betaken rather as an illustrative hypothesis and as an aid in preliminary statementthan as a dogmatic part of the philosophy of physics in its final form.

VII. PRIVATE SPACE AND THE SPACE OF PERSPECTIVES

We have now to explain the ambiguity in the word "place," and how it comesthat two places of different sorts are associated with every sense-datum, namelythe place at which it is and the place from which it is perceived. The theory tobe advocated is closely analogous to Leibniz's monadology, from which itdiffers chiefly in being less smooth and tidy. The first fact to notice is that, so far as can be discovered, no sensibile is evera datum to two people at [159]once. The things seen by two different people areoften closely similar, so similar that the same words can be used to denotethem, without which communication with others concerning sensible objectswould be impossible. But, in spite of this similarity, it would seem that somedifference always arises from difference in the point of view. Thus each person,so far as his sense-data are concerned, lives in a private world. This privateworld contains its own space, or rather spaces, for it would seem that onlyexperience teaches us to correlate the space of sight with the space of touch andwith the various other spaces of other senses. This multiplicity of privatespaces, however, though interesting to the psychologist, is of no greatimportance in regard to our present problem, since a merely solipsisticexperience enables us to correlate them into the one private space whichembraces all our own sense-data. The place at which a sense-datum is, is aplace in private space. This place therefore is different from any place in theprivate space of another percipient. For if we assume, as logical economydemands, that all position is relative, a place is only definable by the things inor around it, and therefore the same place cannot occur in two private worldswhich have no common constituent. The question, therefore, of combiningwhat we call different appearances of the same thing in the same place does notarise, and the fact that a given object appears to different spectators to havedifferent shapes and colours affords no argument against the physical reality ofall these shapes and colours. In addition to the private spaces belonging to the private worlds of differentpercipients, there is, however, another space, in which one whole private worldcounts as a point, or at least as a spatial unit. This might be [160]described as thespace of points of view, since each private world may be regarded as theappearance which the universe presents from a certain point of view. I prefer,however, to speak of it as the space of perspectives, in order to obviate thesuggestion that a private world is only real when someone views it. And for thesame reason, when I wish to speak of a private world without assuming apercipient, I shall call it a "perspective." We have now to explain how the different perspectives are ordered in onespace. This is effected by means of the correlated "sensibilia" which areregarded as the appearances, in different perspectives, of one and the samething. By moving, and by testimony, we discover that two differentperspectives, though they cannot both contain the same "sensibilia," maynevertheless contain very similar ones; and the spatial order of a certain groupof "sensibilia" in a private space of one perspective is found to be identicalwith, or very similar to, the spatial order of the correlated "sensibilia" in theprivate space of another perspective. In this way one "sensibile" in oneperspective is correlated with one "sensibile" in another. Such correlated"sensibilia" will be called "appearances of one thing." In Leibniz's monadology,since each monad mirrored the whole universe, there was in each perspective a"sensibile" which was an appearance of each thing. In our system ofperspectives, we make no such assumption of completeness. A given thing willhave appearances in some perspectives, but presumably not in certain others.The "thing" being defined as the class of its appearances, if κ is the class ofperspectives in which a certain thing θ appears, then θ is a member of themultiplicative class of κ, κ being a class of mutually exclusive classes of"sensibilia." And [161]similarly a perspective is a member of the multiplicativeclass of the things which appear in it. The arrangement of perspectives in a space is effected by means of thedifferences between the appearances of a given thing in the variousperspectives. Suppose, say, that a certain penny appears in a number ofdifferent perspectives; in some it looks larger and in some smaller, in some itlooks circular, in others it presents the appearance of an ellipse of varyingeccentricity. We may collect together all those perspectives in which theappearance of the penny is circular. These we will place on one straight line,ordering them in a series by the variations in the apparent size of the penny.Those perspectives in which the penny appears as a straight line of a certainthickness will similarly be placed upon a plane (though in this case there willbe many different perspectives in which the penny is of the same size; whenone arrangement is completed these will form a circle concentric with thepenny), and ordered as before by the apparent size of the penny. By suchmeans, all those perspectives in which the penny presents a visual appearancecan be arranged in a three-dimensional spatial order. Experience shows that thesame spatial order of perspectives would have resulted if, instead of the penny,we had chosen any other thing which appeared in all the perspectives inquestion, or any other method of utilising the differences between theappearances of the same things in different perspectives. It is this empirical factwhich has made it possible to construct the one all-embracing space of physics. The space whose construction has just been explained, and whose elementsare whole perspectives, will be called "perspective-space."[162]

VIII. THE PLACING OF "THINGS" AND "SENSIBILIA" IN PERSPECTIVE

SPACE

The world which we have so far constructed is a world of six dimensions,

since it is a three-dimensional series of perspectives, each of which is itselfthree-dimensional. We have now to explain the correlation between theperspective space and the various private spaces contained within the variousperspectives severally. It is by means of this correlation that the one three-dimensional space of physics is constructed; and it is because of theunconscious performance of this correlation that the distinction betweenperspective space and the percipient's private space has been blurred, withdisastrous results for the philosophy of physics. Let us revert to our penny: theperspectives in which the penny appears larger are regarded as being nearer tothe penny than those in which it appears smaller, but as far as experience goesthe apparent size of the penny will not grow beyond a certain limit, namely,that where (as we say) the penny is so near the eye that if it were any nearer itcould not be seen. By touch we may prolong the series until the penny touchesthe eye, but no further. If we have been travelling along a line of perspectivesin the previously defined sense, we may, however, by imagining the pennyremoved, prolong the line of perspectives by means, say, of another penny; andthe same may be done with any other line of perspectives defined by means ofthe penny. All these lines meet in a certain place, that is, in a certainperspective. This perspective will be defined as "the place where the penny is." It is now evident in what sense two places in constructed physical space areassociated with a given "sensibile." There is first the place which isthe [163]perspective of which the "sensibile" is a member. This is theplace from which the "sensibile" appears. Secondly there is the place where thething is of which the "sensibile" is a member, in other words an appearance;this is the place at which the "sensibile" appears. The "sensibile" which is amember of one perspective is correlated with another perspective, namely, thatwhich is the place where the thing is of which the "sensibile" is an appearance.To the psychologist the "place from which" is the more interesting, and the"sensibile" accordingly appears to him subjective and where the percipient is.To the physicist the "place at which" is the more interesting, and the "sensibile"accordingly appears to him physical and external. The causes, limits and partialjustification of each of these two apparently incompatible views are evidentfrom the above duplicity of places associated with a given "sensibile." We have seen that we can assign to a physical thing a place in the perspectivespace. In this way different parts of our body acquire positions in perspectivespace, and therefore there is a meaning (whether true or false need not muchconcern us) in saying that the perspective to which our sense-data belong isinside our head. Since our mind is correlated with the perspective to which oursense-data belong, we may regard this perspective as being the position of ourmind in perspective space. If, therefore, this perspective is, in the above definedsense, inside our head, there is a good meaning for the statement that the mindis in the head. We can now say of the various appearances of a given thing thatsome of them are nearer to the thing than others; those are nearer which belongto perspectives that are nearer to "the place where the thing is." We can thusfind a meaning, true or false, for the statement that more is to [164]be learntabout a thing by examining it close to than by viewing it from a distance. Wecan also find a meaning for the phrase "the things which intervene between thesubject and a thing of which an appearance is a datum to him." One reasonoften alleged for the subjectivity of sense-data is that the appearance of a thingmay change when we find it hard to suppose that the thing itself has changed—for example, when the change is due to our shutting our eyes, or to ourscrewing them up so as to make the thing look double. If the thing is defined asthe class of its appearances (which is the definition adopted above), there is ofcourse necessarily some change in the thing whenever any one of itsappearances changes. Nevertheless there is a very important distinctionbetween two different ways in which the appearances may change. If afterlooking at a thing I shut my eyes, the appearance of my eyes changes in everyperspective in which there is such an appearance, whereas most of theappearances of the thing will remain unchanged. We may say, as a matter ofdefinition, that a thing changes when, however near to the thing an appearanceof it may be, there are changes in appearances as near as, or still nearer to, thething. On the other hand we shall say that the change is in some other thing ifall appearances of the thing which are at not more than a certain distance fromthe thing remain unchanged, while only comparatively distant appearances ofthe thing are altered. From this consideration we are naturally led to theconsideration of matter, which must be our next topic.

IX. THE DEFINITION OF MATTER

We defined the "physical thing" as the class of its appearances, but this canhardly be taken as a definition of matter. We want to be able to express the factthat [165]the appearance of a thing in a given perspective is causally affected bythe matter between the thing and the perspective. We have found a meaning for"between a thing and a perspective." But we want matter to be something otherthan the whole class of appearances of a thing, in order to state the influence ofmatter on appearances. We commonly assume that the information we get about a thing is moreaccurate when the thing is nearer. Far off, we see it is a man; then we see it isJones; then we see he is smiling. Complete accuracy would only be attainableas a limit: if the appearances of Jones as we approach him tend towards a limit,that limit may be taken to be what Jones really is. It is obvious that from thepoint of view of physics the appearances of a thing close to "count" more thanthe appearances far off. We may therefore set up the following tentativedefinition: The matter of a given thing is the limit of its appearances as their distancefrom the thing diminishes. It seems probable that there is something in this definition, but it is not quitesatisfactory, because empirically there is no such limit to be obtained fromsense-data. The definition will have to be eked out by constructions anddefinitions. But probably it suggests the right direction in which to look. We are now in a position to understand in outline the reverse journey frommatter to sense-data which is performed by physics. The appearance of a thingin a given perspective is a function of the matter composing the thing and of theintervening matter. The appearance of a thing is altered by intervening smokeor mist, by blue spectacles or by alterations in the sense-organs or nerves of thepercipient (which also must be reckoned as part of the intervening medium).The nearer we approach to[166]the thing, the less its appearance is affected bythe intervening matter. As we travel further and further from the thing, itsappearances diverge more and more from their initial character; and the causallaws of their divergence are to be stated in terms of the matter which liesbetween them and the thing. Since the appearances at very small distances areless affected by causes other than the thing itself, we come to think that thelimit towards which these appearances tend as the distance diminishes is whatthe thing "really is," as opposed to what it merely seems to be. This, togetherwith its necessity for the statement of causal laws, seems to be the source of theentirely erroneous feeling that matter is more "real" than sense-data. Consider for example the infinite divisibility of matter. In looking at a giventhing and approaching it, one sense-datum will become several, and each ofthese will again divide. Thus one appearance may represent many things, and tothis process there seems no end. Hence in the limit, when we approachindefinitely near to the thing there will be an indefinite number of units ofmatter corresponding to what, at a finite distance, is only one appearance. Thisis how infinite divisibility arises. The whole causal efficacy of a thing resides in its matter. This is in somesense an empirical fact, but it would be hard to state it precisely, because"causal efficacy" is difficult to define. What can be known empirically about the matter of a thing is onlyapproximate, because we cannot get to know the appearances of the thing fromvery small distances, and cannot accurately infer the limit of these appearances.But it is inferred approximately by means of the appearances we can observe. Itthen turns out that these appearances can be exhibited by physics as a functionof [167]the matter in our immediate neighbourhood; e.g. the visual appearance ofa distant object is a function of the light-waves that reach the eyes. This leads toconfusions of thought, but offers no real difficulty. One appearance, of a visible object for example, is not sufficient to determineits other simultaneous appearances, although it goes a certain distance towardsdetermining them. The determination of the hidden structure of a thing, so faras it is possible at all, can only be effected by means of elaborate dynamicalinferences.

X. TIME[30]

It seems that the one all-embracing time is a construction, like the one all-embracing space. Physics itself has become conscious of this fact through thediscussions connected with relativity. Between two perspectives which both belong to one person's experience,there will be a direct time-relation of before and after. This suggests a way ofdividing history in the same sort of way as it is divided by differentexperiences, but without introducing experience or any thing mental: we maydefine a "biography" as everything that is (directly) earlier or later than, orsimultaneous with, a given "sensibile." This will give a series of perspectives,which might all form parts of one person's experience, though it is notnecessary that all or any of them should actually do so. By this means, thehistory of the world is divided into a number of mutually exclusive biographies. [168]We have now to correlate the times in the different biographies. Thenatural thing would be to say that the appearances of a given (momentary)thing in two different perspectives belonging to different biographies are to betaken as simultaneous; but this is not convenient. Suppose A shouts to B,and B replies as soon as he hears A'sshout. Then between A's hearing of hisown shout and his hearing of B's there is an interval; thus if wemade A's and B's hearing of the same shout exactly simultaneous with eachother, we should have events exactly simultaneous with a given event but notwith each other. To obviate this, we assume a "velocity of sound." That is, weassume that the time when B hears A's shout is half-way between the timewhen A hears his own shout and the time when he hears B's. In this way thecorrelation is effected. What has been said about sound applies of course equally to light. Thegeneral principle is that the appearances, in different perspectives, which are tobe grouped together as constituting what a certain thing is at a certain moment,are not to be all regarded as being at that moment. On the contrary they spreadoutward from the thing with various velocities according to the nature of theappearances. Since no direct means exist of correlating the time in onebiography with the time in another, this temporal grouping of the appearancesbelonging to a given thing at a given moment is in part conventional. Its motiveis partly to secure the verification of such maxims as that events which areexactly simultaneous with the same event are exactly simultaneous with oneanother, partly to secure convenience in the formulation of causal laws.[169]

XI. THE PERSISTENCE OF THINGS AND MATTER

Apart from any of the fluctuating hypotheses of physics, three main problemsarise in connecting the world of physics with the world of sense, namely:1. the construction of a single space;2. the construction of a single time;3. the construction of permanent things or matter. We have already considered the first and second of these problems; itremains to consider the third. We have seen how correlated appearances in different perspectives arecombined to form one "thing" at one moment in the all-embracing time ofphysics. We have now to consider how appearances at different times arecombined as belonging to one "thing," and how we arrive at the persistent"matter" of physics. The assumption of permanent substance, which technicallyunderlies the procedure of physics, cannot of course be regarded asmetaphysically legitimate: just as the one thing simultaneously seen by manypeople is a construction, so the one thing seen at different times by the same ordifferent people must be a construction, being in fact nothing but a certaingrouping of certain "sensibilia." We have seen that the momentary state of a "thing" is an assemblage of"sensibilia," in different perspectives, not all simultaneous in the oneconstructed time, but spreading out from "the place where the thing is" withvelocities depending upon the nature of the "sensibilia." The time at which the"thing" is in this state is the lower limit of the times at which these appearancesoccur. We have now to consider what leads us to speak of another set ofappearances as belonging to the same "thing" at a different time. [170]For this purpose, we may, at least to begin with, confine ourselves withina single biography. If we can always say when two "sensibilia" in a givenbiography are appearances of one thing, then, since we have seen how toconnect "sensibilia" in different biographies as appearances of the samemomentary state of a thing, we shall have all that is necessary for the completeconstruction of the history of a thing. It is to be observed, to begin with, that the identity of a thing for commonsense is not always correlated with the identity of matter for physics. A humanbody is one persisting thing for common sense, but for physics its matter isconstantly changing. We may say, broadly, that the common-sense conceptionis based upon continuity in appearances at the ordinary distances of sense-data,while the physical conception is based upon the continuity of appearances atvery small distances from the thing. It is probable that the common-senseconception is not capable of complete precision. Let us therefore concentrateour attention upon the conception of the persistence of matter in physics. The first characteristic of two appearances of the same piece of matter atdifferent times is continuity. The two appearances must be connected by aseries of intermediaries, which, if time and space form compact series, mustthemselves form a compact series. The colour of the leaves is different inautumn from what it is in summer; but we believe that the change occursgradually, and that, if the colours are different at two given times, there areintermediate times at which the colours are intermediate between those at thegiven times. But there are two considerations that are important as regards continuity. First, it is largely hypothetical. We do not observe [171]any one thingcontinuously, and it is merely a hypothesis to assume that, while we are notobserving it, it passes through conditions intermediate between those in whichit is perceived. During uninterrupted observation, it is true, continuity is nearlyverified; but even here, when motions are very rapid, as in the case ofexplosions, the continuity is not actually capable of direct verification. Thus wecan only say that the sense-data are found topermit a hypothetical complementof "sensibilia" such as will preserve continuity, and that therefore there may besuch a complement. Since, however, we have already made such use ofhypothetical "sensibilia," we will let this point pass, and admit such"sensibilia," as are required to preserve continuity. Secondly, continuity is not a sufficient criterion of material identity. It is truethat in many cases, such as rocks, mountains, tables, chairs, etc., where theappearances change slowly, continuity is sufficient, but in other cases, such asthe parts of an approximately homogeneous fluid, it fails us utterly. We cantravel by sensibly continuous gradations from any one drop of the sea at anyone time to any other drop at any other time. We infer the motions of sea-waterfrom the effects of the current, but they cannot be inferred from direct sensibleobservation together with the assumption of continuity. The characteristic required in addition to continuity is conformity with thelaws of dynamics. Starting from what common sense regards as persistentthings, and making only such modifications as from time to time seemreasonable, we arrive at assemblages of "sensibilia" which are found to obeycertain simple laws, namely those of dynamics. By regarding "sensibilia" atdifferent times as belonging to the same piece of matter, we are able todefine motion, which presupposes the assumption [172]or construction ofsomething persisting throughout the time of the motion. The motions which areregarded as occurring, during a period in which all the "sensibilia" and thetimes of their appearance are given, will be different according to the manner inwhich we combine "sensibilia" at different times as belonging to the same pieceof matter. Thus even when the whole history of the world is given in everyparticular, the question what motions take place is still to a certain extentarbitrary even after the assumption of continuity. Experience shows that it ispossible to determine motions in such a way as to satisfy the laws of dynamics,and that this determination, roughly and on the whole, is fairly in agreementwith the common-sense opinions about persistent things. This determination,therefore, is adopted, and leads to a criterion by which we can determine,sometimes practically, sometimes only theoretically, whether two appearancesat different times are to be regarded as belonging to the same piece of matter.The persistence of all matter throughout all time can, I imagine, be secured bydefinition. To recommend this conclusion, we must consider what it is that is proved bythe empirical success of physics. What is proved is that its hypotheses, thoughunverifiable where they go beyond sense-data, are at no point in contradictionwith sense-data, but, on the contrary, are ideally such as to render all sense-datacalculable when a sufficient collection of "sensibilia" is given. Now physics hasfound it empirically possible to collect sense-data into series, each series beingregarded as belonging to one "thing," and behaving, with regard to the laws ofphysics, in a way in which series not belonging to one thing would in generalnot behave. If it is to be unambiguous whether two appearances belong to thesame [173]thing or not, there must be only one way of grouping appearances sothat the resulting things obey the laws of physics. It would be very difficult toprove that this is the case, but for our present purposes we may let this pointpass, and assume that there is only one way. Thus we may lay down thefollowing definition: Physical things are those series of appearances whosematter obeys the laws of physics. That such series exist is an empirical fact,which constitutes the verifiability of physics.

XII. ILLUSIONS, HALLUCINATIONS, AND DREAMS

It remains to ask how, in our system, we are to find a place for sense-datawhich apparently fail to have the usual connection with the world of physics.Such sense-data are of various kinds, requiring somewhat different treatment.But all are of the sort that would be called "unreal," and therefore, beforeembarking upon the discussion, certain logical remarks must be made upon theconceptions of reality and unreality. Mr. A. Wolf[31] says: "The conception of mind as a system of transparent activities is, I think, alsountenable because of its failure to account for the very possibility of dreamsand hallucinations. It seems impossible to realise how a bare, transparentactivity can be directed to what is not there, to apprehend what is not given." This statement is one which, probably, most people would endorse. But it isopen to two objections. First it is difficult to see how an activity, however un-"transparent," can be directed towards a nothing: a term of a relation cannot bea mere nonentity. Secondly, no reason [174]is given, and I am convinced thatnone can be given, for the assertion that dream-objects are not "there" and not"given." Let us take the second point first. (1) The belief that dream-objects are not given comes, I think, from failure todistinguish, as regards waking life, between the sense-datum and thecorresponding "thing." In dreams, there is no such corresponding "thing" as thedreamer supposes; if, therefore, the "thing" were given in waking life, as e.g.Meinong maintains,[32] then there would be a difference in respect of givennessbetween dreams and waking life. But if, as we have maintained, what is givenis never the thing, but merely one of the "sensibilia" which compose the thing,then what we apprehend in a dream is just as much given as what we apprehendin waking life. Exactly the same argument applies as to the dream-objects being "there."They have their position in the private space of the perspective of the dreamer;where they fail is in their correlation with other private spaces and thereforewith perspective space. But in the only sense in which "there" can be a datum,they are "there" just as truly as any of the sense-data of waking life. (2) The conception of "illusion" or "unreality," and the correlative conceptionof "reality," are generally used in a way which embodies profound logicalconfusions. Words that go in pairs, such as "real" and "unreal," "existent" and"non-existent," "valid" and "invalid," etc., are all derived from the onefundamental pair, "true" and "false." Now "true" and "false" are applicableonly—except in derivative significations—to propositions. Thus wherever theabove pairs can be significantly applied, we must be dealing either withpropositions or with such incomplete phrases as [175]only acquire meaning whenput into a context which, with them, forms a proposition. Thus such pairs ofwords can be applied to descriptions,[33] but not to proper names: in otherwords, they have no application whatever to data, but only to entities or non-entities described in terms of data. Let us illustrate by the terms "existence" and "non-existence." Given anydatum x, it is meaningless either to assert or to deny that x "exists." We mightbe tempted to say: "Of course x exists, for otherwise it could not be a datum."But such a statement is really meaningless, although it is significant and true tosay "My present sense-datum exists," and it may also be true that "x is mypresent sense-datum." The inference from these two propositions to "x exists" isone which seems irresistible to people unaccustomed to logic; yet the apparentproposition inferred is not merely false, but strictly meaningless. To say "Mypresent sense-datum exists" is to say (roughly): "There is an object of which'my present sense-datum' is a description." But we cannot say: "There is anobject of which 'x' is a description," because 'x' is (in the case we are supposing)a name, not a description. Dr. Whitehead and I have explained this point fullyelsewhere (loc. cit.) with the help of symbols, without which it is hard tounderstand; I shall not therefore here repeat the demonstration of the abovepropositions, but shall proceed with their application to our present problem. The fact that "existence" is only applicable to descriptions is concealed bythe use of what are grammatically proper names in a way which reallytransforms them into descriptions. It is, for example, a legitimate [176]questionwhether Homer existed; but here "Homer" means "the author of the Homericpoems," and is a description. Similarly we may ask whether God exists; butthen "God" means "the Supreme Being" or "the ens realissimum" or whateverother description we may prefer. If "God" were a proper name, God would haveto be a datum; and then no question could arise as to His existence. Thedistinction between existence and other predicates, which Kant obscurely felt,is brought to light by the theory of descriptions, and is seen to remove"existence" altogether from the fundamental notions of metaphysics. What has been said about "existence" applies equally to "reality," which may,in fact, be taken as synonymous with "existence." Concerning the immediateobjects in illusions, hallucinations, and dreams, it is meaningless to ask whetherthey "exist" or are "real." There they are, and that ends the matter. But we maylegitimately inquire as to the existence or reality of "things" or other"sensibilia" inferred from such objects. It is the unreality of these "things" andother "sensibilia," together with a failure to notice that they are not data, whichhas led to the view that the objects of dreams are unreal. We may now apply these considerations in detail to the stock argumentsagainst realism, though what is to be said will be mainly a repetition of whatothers have said before. (1) We have first the variety of normal appearances, supposed to beincompatible. This is the case of the different shapes and colours which a giventhing presents to different spectators. Locke's water which seems both hot andcold belongs to this class of cases. Our system of different perspectives fullyaccounts for these cases, and shows that they afford no argument againstrealism. (2) We have cases where the correlation between [177]different senses isunusual. The bent stick in water belongs here. People say it looks bent but isstraight: this only means that it is straight to the touch, though bent to sight.There is no "illusion," but only a false inference, if we think that the stickwould feel bent to the touch. The stick would look just as bent in a photograph,and, as Mr. Gladstone used to say, "the photograph cannot lie." [34] The case ofseeing double also belongs here, though in this case the cause of the unusualcorrelation is physiological, and would therefore not operate in a photograph. Itis a mistake to ask whether the "thing" is duplicated when we see it double. The"thing" is a whole system of "sensibilia," and it is only those visual "sensibilia"which are data to the percipient that are duplicated. The phenomenon has apurely physiological explanation; indeed, in view of our having two eyes, it isin less need of explanation than the single visual sense-datum which wenormally obtain from the things on which we focus. (3) We come now to cases like dreams, which may, at the moment ofdreaming, contain nothing to arouse suspicion, but are condemned on theground of their supposed incompatibility with earlier and later data. Of courseit often happens that dream-objects fail to behave in the accustomed manner:heavy objects fly, solid objects melt, babies turn into pigs or undergo evengreater changes. But none of these unusual occurrences need happen in adream, and it is not on account of such occurrences that dream-objects arecalled "unreal." It is their lack of continuity with the dreamer's past and futurethat makes him, when he wakes, condemn them; and it is their lack [178]ofcorrelation with other private worlds that makes others condemn them.Omitting the latter ground, our reason for condemning them is that the "things"which we infer from them cannot be combined according to the laws of physicswith the "things" inferred from waking sense-data. This might be used tocondemn the "things" inferred from the data of dreams. Dream-data are nodoubt appearances of "things," but not of such "things" as the dreamersupposes. I have no wish to combat psychological theories of dreams, such asthose of the psycho-analysts. But there certainly are cases where (whateverpsychological causes may contribute) the presence of physical causes also isvery evident. For instance, a door banging may produce a dream of a navalengagement, with images of battleships and sea and smoke. The whole dreamwill be an appearance of the door banging, but owing to the peculiar conditionof the body (especially the brain) during sleep, this appearance is not thatexpected to be produced by a door banging, and thus the dreamer is led toentertain false beliefs. But his sense-data are still physical, and are such as acompleted physics would include and calculate. (4) The last class of illusions are those which cannot be discovered withinone person's experience, except through the discovery of discrepancies with theexperiences of others. Dreams might conceivably belong to this class, if theywere jointed sufficiently neatly into waking life; but the chief instances arerecurrent sensory hallucinations of the kind that lead to insanity. What makesthe patient, in such cases, become what others call insane is the fact that, withinhis own experience, there is nothing to show that the hallucinatory sense-datado not have the usual kind of connection with "sensibilia" in other perspectives.Of course he may learn this through testimony, but he [179]probably finds itsimpler to suppose that the testimony is untrue and that he is being wilfullydeceived. There is, so far as I can see, no theoretical criterion by which thepatient can decide, in such a case, between the two equally satisfactoryhypotheses of his madness and of his friends' mendacity. From the above instances it would appear that abnormal sense-data, of thekind which we regard as deceptive, have intrinsically just the same status asany others, but differ as regards their correlations or causal connections withother "sensibilia" and with "things." Since the usual correlations andconnections become part of our unreflective expectations, and even seem,except to the psychologist, to form part of our data, it comes to be thought,mistakenly, that in such cases the data are unreal, whereas they are merely thecauses of false inferences. The fact that correlations and connections of unusualkinds occur adds to the difficulty of inferring things from sense and ofexpressing physics in terms of sense-data. But the unusualness would seem tobe always physically or physiologically explicable, and therefore raises only acomplication, not a philosophical objection. I conclude, therefore, that no valid objection exists to the view which regardssense-data as part of the actual substance of the physical world, and that, on theother hand, this view is the only one which accounts for the empiricalverifiability of physics. In the present paper, I have given only a roughpreliminary sketch. In particular, the part played by time in the construction ofthe physical world is, I think, more fundamental than would appear from theabove account. I should hope that, with further elaboration, the part played byunperceived "sensibilia" could be indefinitely diminished, probably byinvoking the history of a "thing" to eke out the inferences derivable from itsmomentary appearance.

FOOTNOTES:

[29]Proc. Arist. Soc., 1909-1910, pp. 191-218.

[30]On this subject, compare A Theory of Time and Space, by Mr. A.A. Robb (Camb. Univ. Press),which first suggested to me the views advocated here, though I have, for present purposes, omittedwhat is most interesting and novel in his theory. Mr. Robb has given a sketch of his theory in apamphlet with the same title (Heffer and Sons, Cambridge, 1913).[31]"Natural Realism and Present Tendencies in Philosophy," Proc. Arist. Soc., 1908-1909, p. 165.[32]Die Erfahrungsgrundlagen unseres Wissens, p. 28.[33]Cf. Principia Mathematica, Vol. I, * 14, and Introduction, Chap. III. For the definitionof existence, cf. * 14. 02.[34]Cf. Edwin B. Holt, The Place of Illusory Experience in a Realistic World. "The New Realism," p.303, both on this point and as regards seeing double.

[180]

IXToC

ON THE NOTION OF CAUSE

In the following paper I wish, first, to maintain that the word "cause" is soinextricably bound up with misleading associations as to make its completeextrusion from the philosophical vocabulary desirable; secondly, to inquirewhat principle, if any, is employed in science in place of the supposed "law ofcausality" which philosophers imagine to be employed; thirdly, to exhibitcertain confusions, especially in regard to teleology and determinism, whichappear to me to be connected with erroneous notions as to causality. All philosophers, of every school, imagine that causation is one of thefundamental axioms or postulates of science, yet, oddly enough, in advancedsciences such as gravitational astronomy, the word "cause" never occurs. Dr.James Ward, in his Naturalism and Agnosticism, makes this a ground ofcomplaint against physics: the business of those who wish to ascertain theultimate truth about the world, he apparently thinks, should be the discovery ofcauses, yet physics never even seeks them. To me it seems that philosophyought not to assume such legislative functions, and that the reason why physicshas ceased to look for causes is that, in fact, there are no such things. The lawof causality, I believe, like much that passes muster among philosophers, is arelic of a bygone age, surviving, like the monarchy, only because it iserroneously supposed to do no harm. [181]In order to find out what philosopherscommonly understand by "cause," I consulted Baldwin's Dictionary, and wasrewarded beyond my expectations, for I found the following three mutuallyincompatible definitions:—"CAUSALITY. (1) The necessary connection of events in the time-series...."CAUSE (notion of). Whatever may be included in the thought or perception of a process as taking place in consequence of another process...."CAUSE AND EFFECT. (1) Cause and effect ... are correlative terms denoting any two distinguishable things, phases, or aspects of reality, which are so related to each other that whenever the first ceases to exist the second comes into existence immediately after, and whenever the second comes into existence the first has ceased to exist immediately before." Let us consider these three definitions in turn. The first, obviously, is unintelligible without a definition of "necessary." Under this head, Baldwin's Dictionary gives the following:—"NECESSARY. That is necessary which not only is true, but would be true under all circumstances. Something more than brute compulsion is, therefore, involved in the conception; there is a general law under which the thing takes place." The notion of cause is so intimately connected with that of necessity that it will be no digression to linger over the above definition, with a view to discovering, if possible, some meaning of which it is capable; for, as it stands, it is very far from having any definite signification. The first point to notice is that, if any meaning is to be given to the phrase "would be true under all circumstances," the subject of it must be a propositional [182]function, not a proposition. [35] A proposition is simply true or false, and that ends the matter: there can be no question of "circumstances." "Charles I's head was cut off" is just as true in summer as in winter, on Sundays as on Mondays. Thus when it is worth saying that something "would be true under all circumstances," the something in question must be a propositional function, i.e. an expression containing a variable, and becoming a proposition when a value is assigned to the variable; the varying "circumstances" alluded to are then the different values of which the variable is capable. Thus if "necessary" means "what is true under all circumstances," then "if x is a man, x is mortal" is necessary, because it is true for any possible value of x. Thus we should be led to the following definition:—"NECESSARY is a predicate of a propositional function, meaning that it is true for all possible values of its argument or arguments." Unfortunately, however, the definition in Baldwin's Dictionary says that what is necessary is not only "true under all circumstances" but is also "true." Now these two are incompatible. Only propositions can be "true," and only propositional functions can be "true under all circumstances." Hence the definition as it stands is nonsense. What is meant seems to be this: "Aproposition is necessary when it is a value of a propositional function which istrue under all circumstances, i.e. for all values of its argument or arguments."But if we adopt this definition, the same proposition will be necessary orcontingent according as we choose one or other of its [183]terms as the argumentto our propositional function. For example, "if Socrates is a man, Socrates ismortal," is necessary if Socrates is chosen as argument, but notif man or mortalis chosen. Again, "if Socrates is a man, Plato is mortal," will benecessary if either Socrates or man is chosen as argument, but not if Platoor mortal is chosen. However, this difficulty can be overcome by specifying theconstituent which is to be regarded as argument, and we thus arrive at thefollowing definition: "A proposition is necessary with respect to a given constituent if it remainstrue when that constituent is altered in any way compatible with the propositionremaining significant." We may now apply this definition to the definition of causality quoted above.It is obvious that the argument must be the time at which the earlier eventoccurs. Thus an instance of causality will be such as: "If the event e1 occurs atthe time t1, it will be followed by the event e2." This proposition is intended tobe necessary with respect to t1, i.e. to remain true however t1 may be varied.Causality, as a universal law, will then be the following: "Given any event e1,there is an event e2 such that, whenever e1 occurs, e2 occurs later." But beforethis can be considered precise, we must specify how much later e2 is to occur.Thus the principle becomes:— "Given any event e1, there is an event e2 and a time-interval τ such that,whenever e1 occurs, e2 follows after an interval τ." I am not concerned as yet to consider whether this law is true or false. For thepresent, I am merely concerned to discover what the law of causality issupposed to be. I pass, therefore, to the other definitions quoted above. [184]The second definition need not detain us long, for two reasons. First,because it is psychological: not the "thought or perception" of a process, but theprocess itself, must be what concerns us in considering causality. Secondly,because it is circular: in speaking of a process as "taking place in consequenceof" another process, it introduces the very notion of cause which was to bedefined. The third definition is by far the most precise; indeed as regards clearness itleaves nothing to be desired. But a great difficulty is caused by the temporalcontiguity of cause and effect which the definition asserts. No two instants arecontiguous, since the time-series is compact; hence either the cause or theeffect or both must, if the definition is correct, endure for a finite time; indeed,by the wording of the definition it is plain that both are assumed to endure for afinite time. But then we are faced with a dilemma: if the cause is a processinvolving change within itself, we shall require (if causality is universal) causalrelations between its earlier and later parts; moreover, it would seem that onlythe later parts can be relevant to the effect, since the earlier parts are notcontiguous to the effect, and therefore (by the definition) cannot influence theeffect. Thus we shall be led to diminish the duration of the cause without limit,and however much we may diminish it, there will still remain an earlier partwhich might be altered without altering the effect, so that the true cause, asdefined, will not have been reached, for it will be observed that the definitionexcludes plurality of causes. If, on the other hand, the cause is purely static,involving no change within itself, then, in the first place, no such cause is to befound in nature, and in the second place, it seems strange—too strange to beaccepted, in spite of bare [185]logical possibility—that the cause, after existingplacidly for some time, should suddenly explode into the effect, when it mightjust as well have done so at any earlier time, or have gone on unchangedwithout producing its effect. This dilemma, therefore, is fatal to the view thatcause and effect can be contiguous in time; if there are causes and effects, theymust be separated by a finite time-interval τ, as was assumed in the aboveinterpretation of the first definition. What is essentially the same statement of the law of causality as the oneelicited above from the first of Baldwin's definitions is given by otherphilosophers. Thus John Stuart Mill says:— "The Law of Causation, the recognition of which is the main pillar ofinductive science, is but the familiar truth, that invariability of succession isfound by observation to obtain between every fact in nature and some other factwhich has preceded it."[36] And Bergson, who has rightly perceived that the law as stated byphilosophers is worthless, nevertheless continues to suppose that it is used inscience. Thus he says:— "Now, it is argued, this law [the law of causality] means that everyphenomenon is determined by its conditions, or, in other words, that the samecauses produce the same effects."[37] And again:— "We perceive physical phenomena, and these phenomena obey laws. Thismeans: (1) That phenomena a, b, c, d, previously perceived, can occur again inthe same shape; (2) that a certain phenomenon P, which [186]appeared after theconditions a, b, c, d, and after these conditions only, will not fail to recur assoon as the same conditions are again present."[38] A great part of Bergson's attack on science rests on the assumption that itemploys this principle. In fact, it employs no such principle, but philosophers—even Bergson—are too apt to take their views on science from each other, notfrom science. As to what the principle is, there is a fair consensus amongphilosophers of different schools. There are, however, a number of difficultieswhich at once arise. I omit the question of plurality of causes for the present,since other graver questions have to be considered. Two of these, which areforced on our attention by the above statement of the law, are the following:— (1) What is meant by an "event"? (2) How long may the time-interval be between cause and effect? (1) An "event," in the statement of the law, is obviously intended to besomething that is likely to recur since otherwise the law becomes trivial. Itfollows that an "event" is not a particular, but some universal of which theremay be many instances. It follows also that an "event" must be something shortof the whole state of the universe, since it is highly improbable that this willrecur. What is meant by an "event" is something like striking a match, ordropping a penny into the slot of an automatic machine. If such an event is torecur, it must not be defined too narrowly: we must not state with what degreeof force the match is to be struck, nor what is to be the temperature of thepenny. For if such considerations were relevant, our "event" would occurat [187]most once, and the law would cease to give information. An "event,"then, is a universal defined sufficiently widely to admit of many particularoccurrences in time being instances of it. (2) The next question concerns the time-interval. Philosophers, no doubt,think of cause and effect as contiguous in time, but this, for reasons alreadygiven, is impossible. Hence, since there are no infinitesimal time-intervals,there must be some finite lapse of time τ between cause and effect. This,however, at once raises insuperable difficulties. However short we make theinterval τ, something may happen during this interval which prevents theexpected result. I put my penny in the slot, but before I can draw out my ticketthere is an earthquake which upsets the machine and my calculations. In orderto be sure of the expected effect, we must know that there is nothing in theenvironment to interfere with it. But this means that the supposed cause is not,by itself, adequate to insure the effect. And as soon as we include theenvironment, the probability of repetition is diminished, until at last, when thewhole environment is included, the probability of repetition becomesalmost nil. In spite of these difficulties, it must, of course, be admitted that many fairlydependable regularities of sequence occur in daily life. It is these regularitiesthat have suggested the supposed law of causality; where they are found to fail,it is thought that a better formulation could have been found which would havenever failed. I am far from denying that there may be such sequences which infact never do fail. It may be that there will never be an exception to the rule thatwhen a stone of more than a certain mass, moving with more than a certainvelocity, comes in contact with a pane of glass of [188]less than a certainthickness, the glass breaks. I also do not deny that the observation of suchregularities, even when they are not without exceptions, is useful in the infancyof a science: the observation that unsupported bodies in air usually fall was astage on the way to the law of gravitation. What I deny is that science assumesthe existence of invariable uniformities of sequence of this kind, or that it aimsat discovering them. All such uniformities, as we saw, depend upon a certainvagueness in the definition of the "events." That bodies fall is a vaguequalitative statement; science wishes to know how fast they fall. This dependsupon the shape of the bodies and the density of the air. It is true that there ismore nearly uniformity when they fall in a vacuum; so far as Galileo couldobserve, the uniformity is then complete. But later it appeared that even therethe latitude made a difference, and the altitude. Theoretically, the position ofthe sun and moon must make a difference. In short, every advance in a sciencetakes us farther away from the crude uniformities which are first observed, intogreater differentiation of antecedent and consequent, and into a continuallywider circle of antecedents recognised as relevant. The principle "same cause, same effect," which philosophers imagine to bevital to science, is therefore utterly otiose. As soon as the antecedents havebeen given sufficiently fully to enable the consequent to be calculated withsome exactitude, the antecedents have become so complicated that it is veryunlikely they will ever recur. Hence, if this were the principle involved, sciencewould remain utterly sterile. The importance of these considerations lies partly in the fact that they lead toa more correct account of scientific procedure, partly in the fact that theyremove [189]the analogy with human volition which makes the conception ofcause such a fruitful source of fallacies. The latter point will become clearer bythe help of some illustrations. For this purpose I shall consider a few maximswhich have played a great part in the history of philosophy. (1) "Cause and effect must more or less resemble each other." This principlewas prominent in the philosophy of occasionalism, and is still by no meansextinct. It is still often thought, for example, that mind could not have grown upin a universe which previously contained nothing mental, and one ground forthis belief is that matter is too dissimilar from mind to have been able to causeit. Or, more particularly, what are termed the nobler parts of our nature aresupposed to be inexplicable, unless the universe always contained something atleast equally noble which could cause them. All such views seem to dependupon assuming some unduly simplified law of causality; for, in any legitimatesense of "cause" and "effect," science seems to show that they are usually verywidely dissimilar, the "cause" being, in fact, two states of the whole universe,and the "effect" some particular event. (2) "Cause is analogous to volition, since there must be anintelligible nexus between cause and effect." This maxim is, I think, oftenunconsciously in the imaginations of philosophers who would reject it whenexplicitly stated. It is probably operative in the view we have just beenconsidering, that mind could not have resulted from a purely material world. Ido not profess to know what is meant by "intelligible"; it seems to mean"familiar to imagination." Nothing is less "intelligible," in any other sense, thanthe connection between [190]an act of will and its fulfilment. But obviously thesort of nexus desired between cause and effect is such as could only holdbetween the "events" which the supposed law of causality contemplates; thelaws which replace causality in such a science as physics leave no room for anytwo events between which a nexus could be sought. (3) "The cause compels the effect in some sense in which the effect does notcompel the cause." This belief seems largely operative in the dislike ofdeterminism; but, as a matter of fact, it is connected with our second maxim,and falls as soon as that is abandoned. We may define "compulsion" as follows:"Any set of circumstances is said to compel A when A desires to do somethingwhich the circumstances prevent, or to abstain from something which thecircumstances cause." This presupposes that some meaning has been found forthe word "cause"—a point to which I shall return later. What I want to makeclear at present is that compulsion is a very complex notion, involving thwarteddesire. So long as a person does what he wishes to do, there is no compulsion,however much his wishes may be calculable by the help of earlier events. Andwhere desire does not come in, there can be no question of compulsion. Henceit is, in general, misleading to regard the cause as compelling the effect. A vaguer form of the same maxim substitutes the word "determine" for theword "compel"; we are told that the cause determines the effect in a sense inwhich the effect does not determine the cause. It is not quite clear what ismeant by "determining"; the only precise sense, so far as I know, is that of afunction or one-many relation. If we admit plurality of causes, but not ofeffects, that is, if we suppose that, given the cause, the effect must be such andsuch, but, given the effect, the [191]cause may have been one of manyalternatives, then we may say that the cause determines the effect, but not theeffect the cause. Plurality of causes, however, results only from conceiving theeffect vaguely and narrowly and the cause precisely and widely. Manyantecedents may "cause" a man's death, because his death is vague and narrow.But if we adopt the opposite course, taking as the "cause" the drinking of a doseof arsenic, and as the "effect" the whole state of the world five minutes later,we shall have plurality of effects instead of plurality of causes. Thus thesupposed lack of symmetry between "cause" and "effect" is illusory. (4) "A cause cannot operate when it has ceased to exist, because what hasceased to exist is nothing." This is a common maxim, and a still more commonunexpressed prejudice. It has, I fancy, a good deal to do with the attractivenessof Bergson's "durée": since the past has effects now, it must still exist in somesense. The mistake in this maxim consists in the supposition that causes"operate" at all. A volition "operates" when what it wills takes place; butnothing can operate except a volition. The belief that causes "operate" resultsfrom assimilating them, consciously or unconsciously, to volitions. We havealready seen that, if there are causes at all, they must be separated by a finiteinterval of time from their effects, and thus cause their effects after they haveceased to exist. It may be objected to the above definition of a volition "operating" that itonly operates when it "causes" what it wills, not when it merely happens to befollowed by what it wills. This certainly represents the usual view of what ismeant by a volition "operating," but as it involves the very view of causationwhich we are engaged in combating, it is not open to us as a definition.We [192]may say that a volition "operates" when there is some law in virtue ofwhich a similar volition in rather similar circumstances will usually befollowed by what it wills. But this is a vague conception, and introduces ideaswhich we have not yet considered. What is chiefly important to notice is thatthe usual notion of "operating" is not open to us if we reject, as I contend thatwe should, the usual notion of causation. (5) "A cause cannot operate except where it is." This maxim is verywidespread; it was urged against Newton, and has remained a source ofprejudice against "action at a distance." In philosophy it has led to a denial oftransient action, and thence to monism or Leibnizian monadism. Like theanalogous maxim concerning temporal contiguity, it rests upon the assumptionthat causes "operate," i.e. that they are in some obscure way analogous tovolitions. And, as in the case of temporal contiguity, the inferences drawn fromthis maxim are wholly groundless. I return now to the question, What law or laws can be found to take the placeof the supposed law of causality? First, without passing beyond such uniformities of sequence as arecontemplated by the traditional law, we may admit that, if any such sequencehas been observed in a great many cases, and has never been found to fail, thereis an inductive probability that it will be found to hold in future cases. If stoneshave hitherto been found to break windows, it is probable that they willcontinue to do so. This, of course, assumes the inductive principle, of which thetruth may reasonably be questioned; but as this principle is not our presentconcern, I shall in this discussion treat it as indubitable. We may then say, inthe case of any such frequently observed sequence, that [193]the earlier event isthe cause and the later event the effect. Several considerations, however, make such special sequences very differentfrom the traditional relation of cause and effect. In the first place, the sequence,in any hitherto unobserved instance, is no more than probable, whereas therelation of cause and effect was supposed to be necessary. I do not mean by thismerely that we are not sure of having discovered a true case of cause andeffect; I mean that, even when we have a case of cause and effect in our presentsense, all that is meant is that on grounds of observation, it is probable thatwhen one occurs the other will also occur. Thus in our present sense, A may bethe cause of B even if there actually are cases where B does not follow A.Striking a match will be the cause of its igniting, in spite of the fact that somematches are damp and fail to ignite. In the second place, it will not be assumed that every event has someantecedent which is its cause in this sense; we shall only believe in causalsequences where we find them, without any presumption that they always areto be found. In the third place, any case of sufficiently frequent sequence will be causal inour present sense; for example, we shall not refuse to say that night is the causeof day. Our repugnance to saying this arises from the ease with which we canimagine the sequence to fail, but owing to the fact that cause and effect must beseparated by a finite interval of time, any such sequence might fail through theinterposition of other circumstances in the interval. Mill, discussing thisinstance of night and day, says:— "It is necessary to our using the word cause, that we should believe not onlythat the antecedent always has [194]been followed by the consequent, but that aslong as the present constitution of things endures, it always will be so."[39] In this sense, we shall have to give up the hope of finding causal laws such asMill contemplated; any causal sequence which we have observed may at anymoment be falsified without a falsification of any laws of the kind that the moreadvanced sciences aim at establishing. In the fourth place, such laws of probable sequence, though useful in dailylife and in the infancy of a science, tend to be displaced by quite different lawsas soon as a science is successful. The law of gravitation will illustrate whatoccurs in any advanced science. In the motions of mutually gravitating bodies,there is nothing that can be called a cause, and nothing that can be called aneffect; there is merely a formula. Certain differential equations can be found,which hold at every instant for every particle of the system, and which, giventhe configuration and velocities at one instant, or the configurations at twoinstants, render the configuration at any other earlier or later instanttheoretically calculable. That is to say, the configuration at any instant is afunction of that instant and the configurations at two given instants. Thisstatement holds throughout physics, and not only in the special case ofgravitation. But there is nothing that could be properly called "cause" andnothing that could be properly called "effect" in such a system. No doubt the reason why the old "law of causality" has so long continued topervade the books of philosophers is simply that the idea of a function isunfamiliar to most of them, and therefore they seek an unduly simplifiedstatement. There is no question of repetitions of the "same" cause producing the"same" effect; it [195]is not in any sameness of causes and effects that theconstancy of scientific law consists, but in sameness of relations. And even"sameness of relations" is too simple a phrase; "sameness of differentialequations" is the only correct phrase. It is impossible to state this accurately innon-mathematical language; the nearest approach would be as follows: "Thereis a constant relation between the state of the universe at any instant and therate of change in the rate at which any part of the universe is changing at thatinstant, and this relation is many-one, i.e. such that the rate of change in the rateof change is determinate when the state of the universe is given." If the "law ofcausality" is to be something actually discoverable in the practice of science,the above proposition has a better right to the name than any "law of causality"to be found in the books of philosophers. In regard to the above principle, several observations must be made— (1) No one can pretend that the above principle is a priori or self-evident or a"necessity of thought." Nor is it, in any sense, a premiss of science: it is anempirical generalisation from a number of laws which are themselves empiricalgeneralisations. (2) The law makes no difference between past and future: the future"determines" the past in exactly the same sense in which the past "determines"the future. The word "determine," here, has a purely logical significance: acertain number of variables "determine" another variable if that other variableis a function of them. (3) The law will not be empirically verifiable unless the course of eventswithin some sufficiently small volume [196]will be approximately the same inany two states of the universe which only differ in regard to what is at aconsiderable distance from the small volume in question. For example, motionsof planets in the solar system must be approximately the same however thefixed stars may be distributed, provided that all the fixed stars are very muchfarther from the sun than the planets are. If gravitation varied directly as thedistance, so that the most remote stars made the most difference to the motionsof the planets, the world might be just as regular and just as much subject tomathematical laws as it is at present, but we could never discover the fact. (4) Although the old "law of causality" is not assumed by science, somethingwhich we may call the "uniformity of nature" is assumed, or rather is acceptedon inductive grounds. The uniformity of nature does not assert the trivialprinciple "same cause, same effect," but the principle of the permanence oflaws. That is to say, when a law exhibiting, e.g. an acceleration as a function ofthe configuration has been found to hold throughout the observable past, it isexpected that it will continue to hold in the future, or that, if it does not itselfhold, there is some other law, agreeing with the supposed law as regards thepast, which will hold for the future. The ground of this principle is simply theinductive ground that it has been found to be true in very many instances; hencethe principle cannot be considered certain, but only probable to a degree whichcannot be accurately estimated. The uniformity of nature, in the above sense, although it is assumed in thepractice of science, must not, in its generality, be regarded as a kind of majorpremiss, without which all scientific reasoning would be in error. Theassumption that all laws of nature are permanent has, of [197]course, lessprobability than the assumption that this or that particular law is permanent;and the assumption that a particular law is permanent for all time has lessprobability than the assumption that it will be valid up to such and such a date.Science, in any given case, will assume what the case requires, but no more. Inconstructing the Nautical Almanac for 1915 it will assume that the law ofgravitation will remain true up to the end of that year; but it will make noassumption as to 1916 until it comes to the next volume of the almanac. Thisprocedure is, of course, dictated by the fact that the uniformity of nature is notknown a priori, but is an empirical generalisation, like "all men are mortal." Inall such cases, it is better to argue immediately from the given particularinstances to the new instance, than to argue by way of a major premiss; theconclusion is only probable in either case, but acquires a higher probability bythe former method than by the latter. In all science we have to distinguish two sorts of laws: first, those that areempirically verifiable but probably only approximate; secondly, those that arenot verifiable, but may be exact. The law of gravitation, for example, in itsapplications to the solar system, is only empirically verifiable when it isassumed that matter outside the solar system may be ignored for such purposes;we believe this to be only approximately true, but we cannot empirically verifythe law of universal gravitation which we believe to be exact. This point is veryimportant in connection with what we may call "relatively isolated systems."These may be defined as follows:— A system relatively isolated during a given period is one which, within someassignable margin of error, will behave in the same way throughout that period,however the rest of the universe may be constituted. [198]A system may be called "practically isolated" during a given period if,although there might be states of the rest of the universe which would producemore than the assigned margin of error, there is reason to believe that suchstates do not in fact occur. Strictly speaking, we ought to specify the respect in which the system isrelatively isolated. For example, the earth is relatively isolated as regardsfalling bodies, but not as regards tides; it is practically isolated as regardseconomic phenomena, although, if Jevons' sunspot theory of commercial criseshad been true, it would not have been even practically isolated in this respect. It will be observed that we cannot prove in advance that a system is isolated.This will be inferred from the observed fact that approximate uniformities canbe stated for this system alone. If the complete laws for the whole universewere known, the isolation of a system could be deduced from them; assuming,for example, the law of universal gravitation, the practical isolation of the solarsystem in this respect can be deduced by the help of the fact that there is verylittle matter in its neighbourhood. But it should be observed that isolatedsystems are only important as providing a possibility of discovering scientificlaws; they have no theoretical importance in the finished structure of a science. The case where one event A is said to "cause" another event B, whichphilosophers take as fundamental, is really only the most simplified instance ofa practically isolated system. It may happen that, as a result of general scientificlaws, whenever A occurs throughout a certain period, it is followed by B; inthat case, A and B form a system which is practically isolated throughout thatperiod. It is, however, to be regarded as a piece of good fortune if this occurs; itwill always be due to special[199]circumstances, and would not have been true ifthe rest of the universe had been different though subject to the same laws. The essential function which causality has been supposed to perform is thepossibility of inferring the future from the past, or, more generally, events atany time from events at certain assigned times. Any system in which suchinference is possible may be called a "deterministic" system. We may define adeterministic system as follows:— A system is said to be "deterministic" when, given certain data, e1, e2, ..., en,at times t1, t2, ..., tn respectively, concerning this system, if E_t is the state of thesystem at any time t, there is a functional relation of the form Et = f (e1, t1, e2, t2, ..., en, tn, t). (A) The system will be "deterministic throughout a given period" if t, in theabove formula, may be any time within that period, though outside that periodthe formula may be no longer true. If the universe, as a whole, is such a system,determinism is true of the universe; if not, not. A system which is part of adeterministic system I shall call "determined"; one which is not part of anysuch system I shall call "capricious." The events e1, e2, ..., en I shall call "determinants" of the system. It is to beobserved that a system which has one set of determinants will in general havemany. In the case of the motions of the planets, for example, the configurationsof the solar system at any two given times will be determinants. We may take another illustration from the hypothesis of psycho-physicalparallelism. Let us assume, for the purposes of this illustration, that to a givenstate of brain [200]a given state of mind always corresponds, and vice versa, i.e.that there is a one-one relation between them, so that each is a function of theother. We may also assume, what is practically certain, that to a given state of acertain brain a given state of the whole material universe corresponds, since itis highly improbable that a given brain is ever twice in exactly the same state.Hence there will be a one-one relation between the state of a given person'smind and the state of the whole material universe. It follows that, if n states ofthe material universe are determinants of the material universe, then n states ofa given man's mind are determinants of the whole material and mentaluniverse—assuming, that is to say, that psycho-physical parallelism is true. The above illustration is important in connection with a certain confusionwhich seems to have beset those who have philosophised on the relation ofmind and matter. It is often thought that, if the state of the mind is determinatewhen the state of the brain is given, and if the material world forms adeterministic system, then mind is "subject" to matter in some sense in whichmatter is not "subject" to mind. But if the state of the brain is also determinatewhen the state of the mind is given, it must be exactly as true to regard matteras subject to mind as it would be to regard mind as subject to matter. We could,theoretically, work out the history of mind without ever mentioning matter, andthen, at the end, deduce that matter must meanwhile have gone through thecorresponding history. It is true that if the relation of brain to mind were many-one, not one-one, there would be a one-sided dependence of mind on brain,while conversely, if the relation were one-many, as Bergson supposes, therewould be a one-aided dependence of brain on mind. But the dependenceinvolved is, in any case, only [201]logical; it does not mean that we shall becompelled to do things we desire not to do, which is what people instinctivelyimagine it to mean. As another illustration we may take the case of mechanism and teleology. Asystem may be defined as "mechanical" when it has a set of determinants thatare purely material, such as the positions of certain pieces of matter at certaintimes. It is an open question whether the world of mind and matter, as we knowit, is a mechanical system or not; let us suppose, for the sake of argument, thatit is a mechanical system. This supposition—so I contend—throws no lightwhatever on the question whether the universe is or is not a "teleological"system. It is difficult to define accurately what is meant by a "teleological"system, but the argument is not much affected by the particular definition weadopt. Broadly, a teleological system is one in which purposes are realised, i.e.in which certain desires—those that are deeper or nobler or more fundamentalor more universal or what not—are followed by their realisation. Now thefact—if it be a fact—that the universe is mechanical has no bearing whateveron the question whether it is teleological in the above sense. There might be amechanical system in which all wishes were realised, and there might be one inwhich all wishes were thwarted. The question whether, or how far, our actualworld is teleological, cannot, therefore, be settled by proving that it ismechanical, and the desire that it should be teleological is no ground forwishing it to be not mechanical. There is, in all these questions, a very great difficulty in avoiding confusionbetween what we can infer and what is in fact determined. Let us consider, fora moment, the various senses in which the future may be "determined." Thereis one sense—and a very important [202]one—in which it is determined quiteindependently of scientific laws, namely, the sense that it will be what it willbe. We all regard the past as determined simply by the fact that it has happened;but for the accident that memory works backward and not forward, we shouldregard the future as equally determined by the fact that it will happen. "But,"we are told, "you cannot alter the past, while you can to some extent alter thefuture." This view seems to me to rest upon just those errors in regard tocausation which it has been my object to remove. You cannot make the pastother than it was—true, but this is a mere application of the law ofcontradiction. If you already know what the past was, obviously it is useless towish it different. But also you cannot make the future other than it will be; thisagain is an application of the law of contradiction. And if you happen to knowthe future—e.g. in the case of a forthcoming eclipse—it is just as useless towish it different as to wish the past different. "But," it will be rejoined, "ourwishes can cause the future, sometimes, to be different from what it would be ifthey did not exist, and they can have no such effect upon the past." This, again,is a mere tautology. An effect beingdefined as something subsequent to itscause, obviously we can have no effect upon the past. But that does not meanthat the past would not have been different if our present wishes had beendifferent. Obviously, our present wishes are conditioned by the past, andtherefore could not have been different unless the past had been different;therefore, if our present wishes were different, the past would be different. Ofcourse, the past cannot be different from what it was, but no more can ourpresent wishes be different from what they are; this again is merely the law ofcontradiction. The facts seem to be merely (1) that wishinggenerally [203]depends upon ignorance, and is therefore commoner in regard tothe future than in regard to the past; (2) that where a wish concerns the future, itand its realisation very often form a "practically independent system," i.e. manywishes regarding the future are realised. But there seems no doubt that the maindifference in our feelings arises from the accidental fact that the past but not thefuture can be known by memory. Although the sense of "determined" in which the future is determined by themere fact that it will be what it will be is sufficient (at least so it seems to me)to refute some opponents of determinism, notably M. Bergson and thepragmatists, yet it is not what most people have in mind when they speak of thefuture as determined. What they have in mind is a formula by means of whichthe future can be exhibited, and at least theoretically calculated, as a function ofthe past. But at this point we meet with a great difficulty, which besets what hasbeen said above about deterministic systems, as well as what is said by others. If formulæ of any degree of complexity, however great, are admitted, itwould seem that any system, whose state at a given moment is a function ofcertain measurable quantities, must be a deterministic system. Let us consider,in illustration, a single material particle, whose co-ordinates attime t are xt, yt, zt. Then, however, the particle moves, there must be,theoretically, functions f1, f2, f3, such that xt = ft (t), yt = f2 (t), zt = f3 (t). It follows that, theoretically, the whole state of the material universe attime t must be capable of being exhibited as a function of t. Hence our universewill be deterministic in the sense defined above. But if this be [204]true, noinformation is conveyed about the universe in stating that it is deterministic. Itis true that the formulæ involved may be of strictly infinite complexity, andtherefore not practically capable of being written down or apprehended. Butexcept from the point of view of our knowledge, this might seem to be a detail:in itself, if the above considerations are sound, the material universe must bedeterministic, must be subject to laws. This, however, is plainly not what was intended. The difference between thisview and the view intended may be seen as follows. Given some formula whichfits the facts hitherto—say the law of gravitation—there will be an infinitenumber of other formulæ, not empirically distinguishable from it in the past,but diverging from it more and more in the future. Hence, even assuming thatthere are persistent laws, we shall have no reason for assuming that the law ofthe inverse square will hold in future; it may be some other hithertoindistinguishable law that will hold. We cannot say that every law which hasheld hitherto must hold in the future, because past facts which obey one lawwill also obey others, hitherto indistinguishable but diverging in future. Hencethere must, at every moment, be laws hitherto unbroken which are now brokenfor the first time. What science does, in fact, is to select the simplest formulathat will fit the facts. But this, quite obviously, is merely a methodologicalprecept, not a law of Nature. If the simplest formula ceases, after a time, to beapplicable, the simplest formula that remains applicable is selected, and sciencehas no sense that an axiom has been falsified. We are thus left with the brutefact that, in many departments of science, quite simple laws have hitherto beenfound to hold. This fact cannot be regarded as having any a priori ground, norcan it be used to support inductively the opinion that [205]the same laws willcontinue; for at every moment laws hitherto true are being falsified, though inthe advanced sciences these laws are less simple than those that have remainedtrue. Moreover it would be fallacious to argue inductively from the state of theadvanced sciences to the future state of the others, for it may well be that theadvanced sciences are advanced simply because, hitherto, their subject-matterhas obeyed simple and easily ascertainable laws, while the subject-matter ofother sciences has not done so. The difficulty we have been considering seems to be met partly, if notwholly, by the principle that the time must not enter explicitly into ourformulæ. All mechanical laws exhibit acceleration as a function ofconfiguration, not of configuration and time jointly; and this principle of theirrelevance of the time may be extended to all scientific laws. In fact we mightinterpret the "uniformity of nature" as meaning just this, that no scientific lawinvolves the time as an argument, unless, of course, it is given in an integratedform, in which case lapse of time, though not absolute time, may appear in ourformulæ. Whether this consideration suffices to overcome our difficultycompletely, I do not know; but in any case it does much to diminish it. It will serve to illustrate what has been said if we apply it to the question offree will. (1) Determinism in regard to the will is the doctrine that our volitions belongto some deterministic system, i.e. are "determined" in the sense defined above.Whether this doctrine is true or false, is a mere question of fact; no apriori considerations (if our previous discussions have been correct) can existon either side. On the one hand, there is no a priori category of causality, butmerely certain observed uniformities. As a matter [206]of fact, there are observeduniformities in regard to volitions; thus there is some empirical evidence thatvolitions are determined. But it would be very rash to maintain that theevidence is overwhelming, and it is quite possible that some volitions, as wellas some other things, are not determined, except in the sense in which we foundthat everything must be determined. (2) But, on the other hand, the subjective sense of freedom, sometimesalleged against determinism, has no bearing on the question whatever. Theview that it has a bearing rests upon the belief that causes compel their effects,or that nature enforces obedience to its laws as governments do. These are mereanthropomorphic superstitions, due to assimilation of causes with volitions andof natural laws with human edicts. We feel that our will is not compelled, butthat only means that it is not other than we choose it to be. It is one of thedemerits of the traditional theory of causality that it has created an artificialopposition between determinism and the freedom of which we areintrospectively conscious. (3) Besides the general question whether volitions are determined, there isthe further question whether they are mechanically determined, i.e. whetherthey are part of what was above defined as a mechanical system. This is thequestion whether they form part of a system with purely material determinants,i.e. whether there are laws which, given certain material data, make all volitionsfunctions of those data. Here again, there is empirical evidence up to a point,but it is not conclusive in regard to all volitions. It is important to observe,however that even if volitions are part of a mechanical system, this by nomeans implies any supremacy of matter over mind. It may well be that thesame system which is [207]susceptible of material determinants is alsosusceptible of mental determinants; thus a mechanical system may bedetermined by sets of volitions, as well as by sets of material facts. It wouldseem, therefore, that the reasons which make people dislike the view thatvolitions are mechanically determined are fallacious. (4) The notion of necessity, which is often associated with determinism, is aconfused notion not legitimately deducible from determinism. Three meaningsare commonly confounded when necessity is spoken of:— (α) An action is necessary when it will be performed however much theagent may wish to do otherwise. Determinism does not imply that actions arenecessary in this sense. (β) A propositional function is necessary when all its values are true. Thissense is not relevant to our present discussion. (γ) A proposition is necessary with respect to a given constituent when it isthe value, with that constituent as argument, of a necessary propositionalfunction, in other words, when it remains true however that constituent may bevaried. In this sense, in a deterministic system, the connection of a volition withits determinants is necessary, if the time at which the determinants occur betaken as the constituent to be varied, the time-interval between the determinantsand the volition being kept constant. But this sense of necessity is purelylogical, and has no emotional importance. We may now sum up our discussion of causality. We found first that the lawof causality, as usually stated by philosophers, is false, and is not employed inscience. We then considered the nature of scientific laws, and found that,instead of stating that one event A is always followed [208]by another event B,they stated functional relations between certain events at certain times, whichwe called determinants, and other events at earlier or later times or at the sametime. We were unable to find any a prioricategory involved: the existence ofscientific laws appeared as a purely empirical fact, not necessarily universal,except in a trivial and scientifically useless form. We found that a system withone set of determinants may very likely have other sets of a quite differentkind, that, for example, a mechanically determined system may also beteleologically or volitionally determined. Finally we considered the problem offree will: here we found that the reasons for supposing volitions to bedetermined are strong but not conclusive, and we decided that even if volitionsare mechanically determined, that is no reason for denying freedom in the senserevealed by introspection, or for supposing that mechanical events are notdetermined by volitions. The problem of free will versus determinism istherefore, if we were right, mainly illusory, but in part not yet capable of beingdecisively solved. FOOTNOTES:

[35]A propositional function is an expression containing a variable, or undetermined constituent, and

becoming a proposition as soon as a definite value is assigned to the variable. Examples are: "A is A,""x is a number." The variable is called the argument of the function.[36]Logic, Bk. III, Chap. V, § 2.[37]Time and Free Will, p. 199.[38]Time and Free Will. p. 202.[39]Loc. cit., § 6

[209]

XToC

KNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION

The object of the following paper is to consider what it is that we know in

cases where we know propositions about "the so-and-so" without knowing whoor what the so-and-so is. For example, I know that the candidate who gets mostvotes will be elected, though I do not know who is the candidate who will getmost votes. The problem I wish to consider is: What do we know in thesecases, where the subject is merely described? I have considered this problemelsewhere[40] from a purely logical point of view; but in what follows I wish toconsider the question in relation to theory of knowledge as well as in relation tologic, and in view of the above-mentioned logical discussions, I shall in thispaper make the logical portion as brief as possible. In order to make clear the antithesis between "acquaintance" and"description," I shall first of all try to explain what I mean by "acquaintance." Isay that I amacquainted with an object when I have a direct cognitive relationto that object, i.e. when I am directly aware of the object itself. When I speak ofa cognitive relation here, I do not mean the sort of relation which constitutesjudgment, but the sort which constitutes presentation. In fact, I think therelation of subject and [210]object which I call acquaintance is simply theconverse of the relation of object and subject which constitutes presentation.That is, to say that S has acquaintance with O is essentially the same thing as tosay that O is presented to S. But the associations and natural extensions of theword acquaintance are different from those of the wordpresentation. To beginwith, as in most cognitive words, it is natural to say that I am acquainted withan object even at moments when it is not actually before my mind, provided ithas been before my mind, and will be again whenever occasion arises. This isthe same sense in which I am said to know that 2+2=4 even when I am thinkingof something else. In the second place, the word acquaintance is designed toemphasise, more than the word presentation, the relational character of the factwith which we are concerned. There is, to my mind, a danger that, in speakingof presentation, we may so emphasis the object as to lose sight of the subject.The result of this is either to lead to the view that there is no subject, whencewe arrive at materialism; or to lead to the view that what is presented is part ofthe subject, whence we arrive at idealism, and should arrive at solipsism but forthe most desperate contortions. Now I wish to preserve the dualism of subjectand object in my terminology, because this dualism seems to me a fundamentalfact concerning cognition. Hence I prefer the word acquaintance because itemphasises the need of a subject which is acquainted. When we ask what are the kinds of objects with which we are acquainted, thefirst and most obvious example is sense-data. When I see a colour or hear anoise, I have direct acquaintance with the colour or the noise. The sense-datumwith which I am acquainted in these cases is generally, if not always, complex.This is particularly[211]obvious in the case of sight. I do not mean, of course,merely that the supposed physical object is complex, but that the direct sensibleobject is complex and contains parts with spatial relations. Whether it ispossible to be aware of a complex without being aware of its constituents is notan easy question, but on the whole it would seem that there is no reason why itshould not be possible. This question arises in an acute form in connection withself-consciousness, which we must now briefly consider. In introspection, we seem to be immediately aware of varying complexes,consisting of objects in various cognitive and conative relations to ourselves.When I see the sun, it often happens that I am aware of my seeing the sun, inaddition to being aware of the sun; and when I desire food, it often happens thatI am aware of my desire for food. But it is hard to discover any state of mind inwhich I am aware of myself alone, as opposed to a complex of which I am aconstituent. The question of the nature of self-consciousness is too large andtoo slightly connected with our subject, to be argued at length here. It isdifficult, but probably not impossible, to account for plain facts if we assumethat we do not have acquaintance with ourselves. It is plain that we are notonly acquainted with the complex "Self-acquainted-with-A," but wealso know the proposition "I am acquainted with A." Now here the complex hasbeen analysed, and if "I" does not stand for something which is a direct objectof acquaintance, we shall have to suppose that "I" is something known bydescription. If we wished to maintain the view that there is no acquaintancewith Self, we might argue as follows: We are acquainted with acquaintance,and we know that it is a relation. Also we are acquainted with a complex inwhich we perceive that acquaintance [212]is the relating relation. Hence weknow that this complex must have a constituent which is that which isacquainted, i.e. must have a subject-term as well as an object-term. Thissubject-term we define as "I." Thus "I" means "the subject-term in awarenessesof which I am aware." But as a definition this cannot be regarded as a happyeffort. It would seem necessary, therefore, either to suppose that I amacquainted with myself, and that "I," therefore, requires no definition, beingmerely the proper name of a certain object, or to find some other analysis ofself-consciousness. Thus self-consciousness cannot be regarded as throwinglight on the question whether we can know a complex without knowing itsconstituents. This question, however, is not important for our present purposes,and I shall therefore not discuss it further. The awarenesses we have considered so far have all been awarenesses ofparticular existents, and might all in a large sense be called sense-data. For,from the point of view of theory of knowledge, introspective knowledge isexactly on a level with knowledge derived from sight or hearing. But, inaddition to awareness of the above kind of objects, which may be calledawareness of particulars; we have also (though not quite in the same sense)what may be called awareness of universals. Awareness of universals iscalled conceiving, and a universal of which we are aware is called a concept.Not only are we aware of particular yellows, but if we have seen a sufficientnumber of yellows and have sufficient intelligence, we are aware of theuniversal yellow; this universal is the subject in such judgments as "yellowdiffers from blue" or "yellow resembles blue less than green does." And theuniversal yellow is the predicate in such judgments as "this is yellow," where"this" is a particular sense-datum. And universal relations, too, [213]are objectsof awarenesses; up and down, before and after, resemblance, desire, awarenessitself, and so on, would seem to be all of them objects of which we can beaware. In regard to relations, it might be urged that we are never aware of theuniversal relation itself, but only of complexes in which it is a constituent. Forexample, it may be said that we do not know directly such a relation as before,though we understand such a proposition as "this is before that," and may bedirectly aware of such a complex as "this being before that." This view,however, is difficult to reconcile with the fact that we often know propositionsin which the relation is the subject, or in which the relata are not definite givenobjects, but "anything." For example, we know that if one thing is beforeanother, and the other before a third, then the first is before the third; and herethe things concerned are not definite things, but "anything." It is hard to seehow we could know such a fact about "before" unless we were acquainted with"before," and not merely with actual particular cases of one given object beingbefore another given object. And more directly: A judgment such as "this isbefore that," where this judgment is derived from awareness of a complex,constitutes an analysis, and we should not understand the analysis if we werenot acquainted with the meaning of the terms employed. Thus we must supposethat we are acquainted with the meaning of "before," and not merely withinstances of it. There are thus at least two sorts of objects of which we are aware, namely,particulars and universals. Among particulars I include all existents, and allcomplexes of which one or more constituents are existents, such as this-before-that, this-above-that, the-yellowness-of-this. [214]Among universals I include allobjects of which no particular is a constituent. Thus the disjunction "universal-particular" includes all objects. We might also call it the disjunction "abstract-concrete." It is not quite parallel with the opposition "concept-percept," becausethings remembered or imagined belong with particulars, but can hardly becalled percepts. (On the other hand, universals with which we are acquaintedmay be identified with concepts.) It will be seen that among the objects with which we are acquainted are notincluded physical objects (as opposed to sense-data), nor other people's minds.These things are known to us by what I call "knowledge by description," whichwe must now consider. By a "description" I mean any phrase of the form "a so-and-so" or "the so-and-so." A phrase of the form "a so-and-so" I shall call an "ambiguous"description; a phrase of the form "the so-and-so" (in the singular) I shall call a"definite" description. Thus "a man" is an ambiguous description, and "the manwith the iron mask" is a definite description. There are various problemsconnected with ambiguous descriptions, but I pass them by, since they do notdirectly concern the matter I wish to discuss. What I wish to discuss is thenature of our knowledge concerning objects in cases where we know that thereis an object answering to a definite description, though we arenot acquainted with any such object. This is a matter which is concernedexclusively with definite descriptions. I shall, therefore, in the sequel, speaksimply of "descriptions" when I mean "definite descriptions." Thus adescription will mean any phrase of the form "the so-and-so" in the singular. I shall say that an object is "known by description" when we know that it is"the so-and-so," i.e. when we [215]know that there is one object, and no more,having a certain property; and it will generally be implied that we do not haveknowledge of the same object by acquaintance. We know that the man with theiron mask existed, and many propositions are known about him; but we do notknow who he was. We know that the candidate who gets most votes will beelected, and in this case we are very likely also acquainted (in the only sense inwhich one can be acquainted with some one else) with the man who is, in fact,the candidate who will get most votes, but we do not know which of thecandidates he is, i.e. we do not know any proposition of the form "A is thecandidate who will get most votes" where A is one of the candidates by name.We shall say that we have "merely descriptive knowledge" of the so-and-sowhen, although we know that the so-and-so exists, and although we maypossibly be acquainted with the object which is, in fact, the so-and-so, yet wedo not know any proposition "a is the so-and-so," where a is something withwhich we are acquainted. When we say "the so-and-so exists," we mean that there is just one objectwhich is the so-and-so. The proposition "a is the so-and-so" means that a hasthe property so-and-so, and nothing else has. "Sir Joseph Larmor is theUnionist candidate" means "Sir Joseph Larmor is a Unionist candidate, and noone else is." "The Unionist candidate exists" means "some one is a Unionistcandidate, and no one else is." Thus, when we are acquainted with an objectwhich we know to be the so-and-so, we know that the so-and-so exists but wemay know that the so-and-so exists when we are not acquainted with any objectwhich we know to be the so-and-so, and even when we are not acquainted withany object which, in fact, is the so-and-so. [216]Common words, even proper names, are usually really descriptions. Thatis to say, the thought in the mind of a person using a proper name correctly cangenerally only be expressed explicitly if we replace the proper name by adescription. Moreover, the description required to express the thought will varyfor different people, or for the same person at different times. The only thingconstant (so long as the name is rightly used) is the object to which the nameapplies. But so long as this remains constant, the particular description involvedusually makes no difference to the truth or falsehood of the proposition inwhich the name appears. Let us take some illustrations. Suppose some statement made aboutBismarck. Assuming that there is such a thing as direct acquaintance withoneself, Bismarck himself might have used his name directly to designate theparticular person with whom he was acquainted. In this case, if he made ajudgment about himself, he himself might be a constituent of the judgment.Here the proper name has the direct use which it always wishes to have, assimply standing for a certain object, and not for a description of the object. Butif a person who knew Bismarck made a judgment about him, the case isdifferent. What this person was acquainted with were certain sense-data whichhe connected (rightly, we will suppose) with Bismarck's body. His body as aphysical object, and still more his mind, were only known as the body and themind connected with these sense-data. That is, they were known by description.It is, of course, very much a matter of chance which characteristics of a man'sappearance will come into a friend's mind when he thinks of him; thus thedescription actually in the friend's mind is accidental. The essential point is thathe knows that the various descriptions all apply to the [217]same entity, in spiteof not being acquainted with the entity in question. When we, who did not know Bismarck, make a judgment about him, thedescription in our minds will probably be some more or less vague mass ofhistorical knowledge—far more, in most cases, than is required to identify him.But, for the sake of illustration, let us assume that we think of him as "the firstChancellor of the German Empire." Here all the words are abstract except"German." The word "German" will again have different meanings for differentpeople. To some it will recall travels in Germany, to some the look of Germanyon the map, and so on. But if we are to obtain a description which we know tobe applicable, we shall be compelled, at some point, to bring in a reference to aparticular with which we are acquainted. Such reference is involved in anymention of past, present, and future (as opposed to definite dates), or of hereand there, or of what others have told us. Thus it would seem that, in some wayor other, a description known to be applicable to a particular must involve somereference to a particular with which we are acquainted, if our knowledge aboutthe thing described is not to be merely what follows logically from thedescription. For example, "the most long-lived of men" is a description whichmust apply to some man, but we can make no judgments concerning this manwhich involve knowledge about him beyond what the description gives. If,however, we say, "the first Chancellor of the German Empire was an astutediplomatist," we can only be assured of the truth of our judgment in virtue ofsomething with which we are acquainted—usually a testimony heard or read.Considered psychologically, apart from the information we convey to others,apart from the fact about the actual [218]Bismarck, which gives importance toour judgment, the thought we really have contains the one or more particularsinvolved, and otherwise consists wholly of concepts. All names of places—London, England, Europe, the earth, the Solar System—similarly involve,when used, descriptions which start from some one or more particulars withwhich we are acquainted. I suspect that even the Universe, as considered bymetaphysics, involves such a connection with particulars. In logic, on thecontrary, where we are concerned not merely with what does exist, but withwhatever might or could exist or be, no reference to actual particulars isinvolved. It would seem that, when we make a statement about something only knownby description, we often intend to make our statement, not in the forminvolving the description, but about the actual thing described. That is to say,when we say anything about Bismarck, we should like, if we could, to make thejudgment which Bismarck alone can make, namely, the judgment of which hehimself is a constituent. In this we are necessarily defeated, since the actualBismarck is unknown to us. But we know that there is an object B calledBismarck, and that B was an astute diplomatist. We can thus describe theproposition we should like to affirm, namely, "B was an astute diplomatist,"where B is the object which was Bismarck. What enables us to communicate inspite of the varying descriptions we employ is that we know there is a trueproposition concerning the actual Bismarck, and that, however we may vary thedescription (so long as the description is correct), the proposition described isstill the same. This proposition, which is described and is known to be true, iswhat interests us; but we are not acquainted with the proposition itself, and donot knowit, though we know it is true. [219]Itwill be seen that there are various stages in the removal fromacquaintance with particulars: there is Bismarck to people who knew him,Bismarck to those who only know of him through history, the man with theiron mask, the longest-lived of men. These are progressively further removedfrom acquaintance with particulars, and there is a similar hierarchy in theregion of universals. Many universals, like many particulars, are only known tous by description. But here, as in the case of particulars, knowledge concerningwhat is known by description is ultimately reducible to knowledge concerningwhat is known by acquaintance. The fundamental epistemological principle in the analysis of propositionscontaining descriptions is this: Every proposition which we can understandmust be composed wholly of constituents with which we are acquainted. Fromwhat has been said already, it will be plain why I advocate this principle, andhow I propose to meet the case of propositions which at first sight contraveneit. Let us begin with the reasons for supposing the principle true. The chief reason for supposing the principle true is that it seems scarcelypossible to believe that we can make a judgment or entertain a suppositionwithout knowing what it is that we are judging or supposing about. If we makea judgment about (say) Julius Cæsar, it is plain that the actual person who wasJulius Cæsar is not a constituent of the judgment. But before going further, itmay be well to explain what I mean when I say that this or that is a constituentof a judgment, or of a proposition which we understand. To begin withjudgments: a judgment, as an occurrence, I take to be a relation of a mind toseveral entities, namely, the entities which compose what is judged. If, e.g. Ijudge [220]that A loves B, the judgment as an event consists in the existence, at acertain moment, of a specific four-term relation, called judging, between meand A and love and B. That is to say, at the time when I judge, there is a certaincomplex whose terms are myself and A and love and B, and whose relatingrelation is judging. My reasons for this view have been set forthelsewhere,[41] and I shall not repeat them here. Assuming this view ofjudgment, the constituents of the judgment are simply the constituents of thecomplex which is the judgment. Thus, in the above case, the constituents aremyself and A and love and B and judging. But myself and judging areconstituents shared by all my judgments; thus the distinctive constituents of theparticular judgment in question are A and love and B. Coming now to what ismeant by "understanding a proposition," I should say that there is anotherrelation possible between me and A and love and B, which is calledmysupposing that A loves B.[42] When we can suppose that A loves B, we"understand the proposition" A loves B. Thus we often understand a propositionin cases where we have not enough knowledge to make a judgment. Supposing,like judging, is a many-term relation, of which a mind is one term. The otherterms of the relation are called the constituents of the proposition supposed.Thus the principle which I enunciated may be re-stated as follows: Whenevera [221]relation of supposing or judging occurs, the terms to which the supposingor judging mind is related by the relation of supposing or judging must beterms with which the mind in question is acquainted. This is merely to say thatwe cannot make a judgment or a supposition without knowing what it is that weare making our judgment or supposition about. It seems to me that the truth ofthis principle is evident as soon as the principle is understood; I shall, therefore,in what follows, assume the principle, and use it as a guide in analysingjudgments that contain descriptions. Returning now to Julius Cæsar, I assume that it will be admitted that hehimself is not a constituent of any judgment which I can make. But at this pointit is necessary to examine the view that judgments are composed of somethingcalled "ideas," and that it is the "idea" of Julius Cæsar that is a constituent ofmy judgment. I believe the plausibility of this view rests upon a failure to forma right theory of descriptions. We may mean by my "idea" of Julius Cæsar thethings that I know about him, e.g. that he conquered Gaul, was assassinated onthe Ides of March, and is a plague to schoolboys. Now I am admitting, andindeed contending, that in order to discover what is actually in my mind when Ijudge about Julius Cæsar, we must substitute for the proper name a descriptionmade up of some of the things I know about him. (A description which willoften serve to express my thought is "the man whose name was Julius Cæsar."For whatever else I may have forgotten about him, it is plain that when Imention him I have not forgotten that that was his name.) But although I thinkthe theory that judgments consist of ideas may have been suggested in somesuch way, yet I think the theory itself is fundamentally mistaken. The [222]viewseems to be that there is some mental existent which may be called the "idea"of something outside the mind of the person who has the idea, and that, sincejudgment is a mental event, its constituents must be constituents of the mind ofthe person judging. But in this view ideas become a veil between us and outsidethings—we never really, in knowledge, attain to the things we are supposed tobe knowing about, but only to the ideas of those things. The relation of mind,idea, and object, on this view, is utterly obscure, and, so far as I can see,nothing discoverable by inspection warrants the intrusion of the idea betweenthe mind and the object. I suspect that the view is fostered by the dislike ofrelations, and that it is felt the mind could not know objects unless there weresomething "in" the mind which could be called the state of knowing the object.Such a view, however, leads at once to a vicious endless regress, since therelation of idea to object will have to be explained by supposing that the ideaitself has an idea of the object, and so on ad infinitum. I therefore see no reasonto believe that, when we are acquainted with an object, there is in us somethingwhich can be called the "idea" of the object. On the contrary, I hold thatacquaintance is wholly a relation, not demanding any such constituent of themind as is supposed by advocates of "ideas." This is, of course, a largequestion, and one which would take us far from our subject if it wereadequately discussed. I therefore content myself with the above indications, andwith the corollary that, in judging, the actual objects concerning which wejudge, rather than any supposed purely mental entities, are constituents of thecomplex which is the judgment. When, therefore, I say that we must substitute for "Julius Cæsar" somedescription of Julius Cæsar, in order [223]to discover the meaning of a judgmentnominally about him, I am not saying that we must substitute an idea. Supposeour description is "the man whose name was Julius Cæsar." Let our judgmentbe "Julius Cæsar was assassinated." Then it becomes "the man whose namewas Julius Cæsar was assassinated." Here Julius Cæsar is a noise or shape withwhich we are acquainted, and all the other constituents of the judgment(neglecting the tense in "was") are concepts with which we are acquainted.Thus our judgment is wholly reduced to constituents with which we areacquainted, but Julius Cæsar himself has ceased to be a constituent of ourjudgment. This, however, requires a proviso, to be further explained shortly,namely that "the man whose name was Julius Cæsar" must not, as a whole, bea constituent of our judgment, that is to say, this phrase must not, as a whole,have a meaning which enters into the judgment. Any right analysis of thejudgment, therefore, must break up this phrase, and not treat it as a subordinatecomplex which is part of the judgment. The judgment "the man whose namewas Julius Cæsar was assassinated" may be interpreted as meaning "one andonly one man was calledJulius Cæsar, and that one was assassinated." Here itis plain that there is no constituent corresponding to the phrase "the man whosename was Julius Cæsar." Thus there is no reason to regard this phrase asexpressing a constituent of the judgment, and we have seen that this phrasemust be broken up if we are to be acquainted with all the constituents of thejudgment. This conclusion, which we have reached from considerationsconcerned with the theory of knowledge, is also forced upon us by logicalconsiderations, which must now be briefly reviewed. It is common to distinguish two aspects, meaning and [224]denotation, suchphrases as "the author of Waverley." The meaning will be a certain complex,consisting (at least) of authorship and Waverley with some relation; thedenotation will be Scott. Similarly "featherless bipeds" will have a complexmeaning, containing as constituents the presence of two feet and the absence offeathers, while its denotation will be the class of men. Thus when we say "Scottis the author of Waverley" or "men are the same as featherless bipeds," we areasserting an identity of denotation, and this assertion is worth making becauseof the diversity of meaning.[43] I believe that the duality of meaning anddenotation, though capable of a true interpretation, is misleading if taken asfundamental. The denotation, I believe, is not a constituent of the proposition,except in the case of proper names, i.e. of words which do not assign a propertyto an object, but merely and solely name it. And I should hold further that, inthis sense, there are only two words which are strictly proper names ofparticulars, namely, "I" and "this." [44] One reason for not believing the denotation to be a constituent of theproposition is that we may know the proposition even when we are notacquainted with the denotation. The proposition "the author of Waverley is anovelist" was known to people who did not know that "the author of Waverley"denoted Scott. This reason has been already sufficiently emphasised. A second reason is that propositions concerning "the so-and-so" are possibleeven when "the so-and-so" has no denotation. Take, e.g. "the golden mountaindoes not exist" or "the round square is self-contradictory." [225]If we are topreserve the duality of meaning and denotation, we have to say, with Meinong,that there are such objects as the golden mountain and the round square,although these objects do not have being. We even have to admit that theexistent round square is existent, but does not exist. [45] Meinong does notregard this as a contradiction, but I fail to see that it is not one. Indeed, it seemsto me evident that the judgment "there is no such object as the round square"does not presuppose that there is such an object. If this is admitted, however,we are led to the conclusion that, by parity of form, no judgment concerning"the so-and-so" actually involves the so-and-so as a constituent. Miss Jones[46] contends that there is no difficulty in admitting contradictorypredicates concerning such an object as "the present King of France," on theground that this object is in itself contradictory. Now it might, of course, beargued that this object, unlike the round square, is not self-contradictory, butmerely non-existent. This, however, would not go to the root of the matter. Thereal objection to such an argument is that the law of contradiction ought not tobe stated in the traditional form "A is not both B and not B," but in the form"no proposition is both true and false." The traditional form only applies tocertain propositions, namely, to those which attribute a predicate to a subject.When the law is stated of propositions, instead of being stated concerningsubjects and predicates, it is at once evident that propositions about the presentKing of France or the round square can form no exception, but are just asincapable of being both true and false as other propositions. MissJones[47] argues that "Scott is the author of [226]Waverley" asserts identity ofdenotation between Scott and the author of Waverley. But there is somedifficulty in choosing among alternative meanings of this contention. In thefirst place, it should be observed that the author of Waverley is nota mere name, like Scott. Scott is merely a noise or shape conventionally used todesignate a certain person; it gives us no information about that person, and hasnothing that can be called meaning as opposed to denotation. (I neglect the fact,considered above, that even proper names, as a rule, really stand fordescriptions.) But the author of Waverley is not merely conventionally a namefor Scott; the element of mere convention belongs here to the separatewords, the and author and of and Waverley. Given what these words standfor, the author of Waverley is no longer arbitrary. When it is said that Scott isthe author of Waverley, we are not stating that these are two names for oneman, as we should be if we said "Scott is Sir Walter." A man's name is what heis called, but however much Scott had been called the author of Waverley, thatwould not have made him be the author; it was necessary for him actually towrite Waverley, which was a fact having nothing to do with names. If, then, we are asserting identity of denotation, we must not meanby denotation the mere relation of a name to the thing named. In fact, it wouldbe nearer to the truth to say that the meaning of "Scott" is the denotation of "theauthor of Waverley." The relation of "Scott" to Scott is that "Scott" meansScott, just as the relation of "author" to the concept which is so called is that"author" means this concept. Thus if we distinguish meaning and denotation in"the author of Waverley," we shall have to say that "Scott" has meaning but notdenotation. Also when we say "Scott is the [227]author of Waverley,"the meaning of "the author of Waverley" is relevant to our assertion. For if thedenotation alone were relevant, any other phrase with the same denotationwould give the same proposition. Thus "Scott is the author of Marmion" wouldbe the same proposition as "Scott is the author of Waverley." But this is plainlynot the case, since from the first we learn that Scott wrote Marmion and fromthe second we learn that he wrote Waverley, but the first tells us nothing aboutWaverley and the second nothing about Marmion. Hence the meaning of "theauthor of Waverley," as opposed to the denotation, is certainly relevant to"Scott is the author of Waverley." We have thus agreed that "the author of Waverley" is not a mere name, andthat its meaning is relevant in propositions in which it occurs. Thus if we are tosay, as Miss Jones does, that "Scott is the author of Waverley" asserts anidentity of denotation, we must regard the denotation of "the author ofWaverley" as the denotation of what is meant by "the author of Waverley." Letus call the meaning of "the author of Waverley" M. Thus M is what "the authorof Waverley" means. Then we are to suppose that "Scott is the author ofWaverley" means "Scott is the denotation of M." But here we are explainingour proposition by another of the same form, and thus we have made noprogress towards a real explanation. "The denotation of M," like "the author ofWaverley," has both meaning and denotation, on the theory we are examining.If we call its meaning M', our proposition becomes "Scott is the denotation ofM'." But this leads at once to an endless regress. Thus the attempt to regard ourproposition as asserting identity of denotation breaks down, and it becomesimperative to find some other analysis. When this analysis hasbeen [228]completed, we shall be able to reinterpret the phrase "identity ofdenotation," which remains obscure so long as it is taken as fundamental. The first point to observe is that, in any proposition about "the author ofWaverley," provided Scott is not explicitly mentioned, the denotation itself, i.e.Scott, does not occur, but only the concept of denotation, which will berepresented by a variable. Suppose we say "the author of Waverley was theauthor of Marmion," we are certainly not saying that both were Scott—we mayhave forgotten that there was such a person as Scott. We are saying that there issome man who was the author of Waverley and the author of Marmion. That isto say, there is some one who wrote Waverley and Marmion, and no one elsewrote them. Thus the identity is that of a variable, i.e. of an indefinite subject,"some one." This is why we can understand propositions about "the author ofWaverley," without knowing who he was. When we say "the author ofWaverley was a poet," we mean "one and only one man wrote Waverley, andhe was a poet"; when we say "the author of Waverley was Scott" we mean "oneand only one man wrote Waverley, and he was Scott." Here the identity isbetween a variable, i.e. an indeterminate subject ("he"), and Scott; "the authorof Waverley" has been analysed away, and no longer appears as a constituentof the proposition.[48] The reason why it is imperative to analyse away the phrase "the author ofWaverley" may be stated as follows. It is plain that when we say "the author ofWaverley is the author of Marmion," the is expresses [229]identity. We haveseen also that the common denotation, namely Scott, is not a constituent of thisproposition, while themeanings (if any) of "the author of Waverley" and "theauthor of Marmion" are not identical. We have seen also that, in any sense inwhich the meaning of a word is a constituent of a proposition in whose verbalexpression the word occurs, "Scott" means the actual man Scott, in the samesense (so far as concerns our present discussion) in which "author" means acertain universal. Thus, if "the author of Waverley" were a subordinatecomplex in the above proposition, its meaning would have to be what was saidto be identical with the meaning of "the author of Marmion." This is plainly notthe case; and the only escape is to say that "the author of Waverley" does not,by itself, have a meaning, though phrases of which it is part do have a meaning.That is, in a right analysis of the above proposition, "the author of Waverley"must disappear. This is effected when the above proposition is analysed asmeaning: "Some one wrote Waverley and no one else did, and that some onealso wrote Marmion and no one else did." This may be more simply expressedby saying that the propositional function "x wrote Waverley and Marmion, andno one else did" is capable of truth, i.e. some value of x makes it true, but noother value does. Thus the true subject of our judgment is a propositionalfunction, i.e. a complex containing an undetermined constituent, and becominga proposition as soon as this constituent is determined. We may now define the denotation of a phrase. If we know that theproposition "a is the so-and-so" is true, i.e. that a is so-and-so and nothing elseis, we call a the denotation of the phrase "the so-and-so." A very great many ofthe propositions we naturally make about "the [230]so-and-so" will remain trueor remain false if we substitutea for "the so-and-so," where a is the denotationof "the so-and-so." Such propositions will also remain true or remain false if wesubstitute for "the so-and-so" any other phrase having the same denotation.Hence, as practical men, we become interested in the denotation more than inthe description, since the denotation decides as to the truth or falsehood of somany statements in which the description occurs. Moreover, as we saw earlierin considering the relations of description and acquaintance, we often wish toreach the denotation, and are only hindered by lack of acquaintance: in suchcases the description is merely the means we employ to get as near as possibleto the denotation. Hence it naturally comes to be supposed that the denotation ispart of the proposition in which the description occurs. But we have seen, bothon logical and on epistemological grounds, that this is an error. The actualobject (if any) which is the denotation is not (unless it is explicitly mentioned) aconstituent of propositions in which descriptions occur; and this is the reasonwhy, in order to understand such propositions, we need acquaintance with theconstituents of the description, but do not need acquaintance with itsdenotation. The first result of analysis, when applied to propositions whosegrammatical subject is "the so-and-so," is to substitute a variable as subject; i.e.we obtain a proposition of the form: "There is something which alone is so-and-so, and that something is such-and-such." The further analysis of propositionsconcerning "the so-and-so" is thus merged in the problem of the nature of thevariable, i.e. of the meanings of some, any, and all. This is a difficult problem,concerning which I do not intend to say anything at present. To sum up our whole discussion. We began by [231]distinguishing two sorts ofknowledge of objects, namely, knowledge by acquaintance and knowledgebydescription. Of these it is only the former that brings the object itself beforethe mind. We have acquaintance with sense-data, with many universals, andpossibly with ourselves, but not with physical objects or other minds. Wehave descriptive knowledge of an object when we know that it is the objecthaving some property or properties with which we are acquainted; that is tosay, when we know that the property or properties in question belong to oneobject and no more, we are said to have knowledge of that one object bydescription, whether or not we are acquainted with the object. Our knowledgeof physical objects and of other minds is only knowledge by description, thedescriptions involved being usually such as involve sense-data. All propositionsintelligible to us, whether or not they primarily concern things only known tous by description, are composed wholly of constituents with which we areacquainted, for a constituent with which we are not acquainted is unintelligibleto us. A judgment, we found, is not composed of mental constituents called"ideas," but consists of an occurrence whose constituents are a mind [49] andcertain objects, particulars or universals. (One at least must be a universal.)When a judgment is rightly analysed, the objects which are constituents of itmust all be objects with which the mind which is a constituent of it isacquainted. This conclusion forces us to analyse descriptive phrases occurringin propositions, and to say that the objects denoted by such phrases are notconstituents of judgments in which such phrases occur (unless these objects areexplicitly [232]mentioned). This leads us to the view (recommended also onpurely logical grounds) that when we say "the author of Marmion was theauthor of Waverley," Scott himself is not a constituent of our judgment, andthat the judgment cannot be explained by saying that it affirms identity ofdenotation with diversity of meaning. It also, plainly, does not assert identity ofmeaning. Such judgments, therefore, can only be analysed by breaking up thedescriptive phrases, introducing a variable, and making propositional functionsthe ultimate subjects. In fact, "the so-and-so is such-and-such" will mean that"x is so-and-so and nothing else is, and x is such-and-such" is capable of truth.The analysis of such judgments involves many fresh problems, but thediscussion of these problems is not undertaken in the present paper.

FOOTNOTES:

[40]See references later.

[41]Philosophical Essays, "The Nature of Truth." I have been persuaded by Mr. Wittgenstein that thistheory is somewhat unduly simple, but the modification which I believe it to require does not affectthe above argument [1917].[42]Cf. Meinong, Ueber Annahmen, passim. I formerly supposed, contrary to Meinong's view, that therelationship of supposing might be merely that of presentation. In this view I now think I wasmistaken, and Meinong is right. But my present view depends upon the theory that both in judgmentand in assumption there is no single Objective, but the several constituents of the judgment orassumption are in a many-term relation to the mind.[43]This view has been recently advocated by Miss E.E.C. Jones. "A New Law of Thought and itsImplications," Mind, January, 1911.[44]I should now exclude "I" from proper names in the strict sense, and retain only "this" [1917].[45]Meinong, Ueber Annahmen, 2nd ed., Leipzig, 1910, p. 141.[46]Mind, July, 1910, p. 380.[47]Mind, July, 1910, p. 379.[48]The theory which I am advocating is set forth fully, with the logical grounds in its favour,in Principia Mathematica, Vol. I. Introduction, Chap. III; also, less fully, in Mind, October, 1905.[49]I use this phrase merely to denote the something psychological which enters into judgment,without intending to prejudge the question as to what this something is.[233]

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